Avogadro's Number and Gram Mole
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Questions and Answers

What is the value of Avogadro's number?

  • 6.024 × 10^22
  • 6.024 × 10^23 (correct)
  • 6.024 × 10^25
  • 6.024 × 10^24

    • How many atoms of oxygen are in 16 grams of oxygen?

  • 6.024 × 10^24
  • 6.024 × 10^23 (correct)
  • 6.024 × 10^25
  • 6.024 × 10^22

    • What is the weight in grams of one gram mole of sulfur?

  • 128 grams
  • 64 grams
  • 32 grams (correct)
  • 16 grams

    • How many gram moles of sulfur are in 64 kilograms of sulfur?

    <p>2000</p> Signup and view all the answers

    How many atoms of sulfur are in 64 kilograms of sulfur?

    <p>12.048 × 10^26</p> Signup and view all the answers

    What is the weight in grams of one atom of sodium?

    <p>23 / 6.024 × 10^23</p> Signup and view all the answers

    If one gram mole of an element has a weight of 40 grams, how many atoms are in 80 grams of the element?

    <p>2 × 6.024 × 10^23</p> Signup and view all the answers

    What is the units of Avogadro's number?

    <p>atoms per gram mole</p> Signup and view all the answers

    What is the formula for calculating the weight of an object?

    <p>W = mg</p> Signup and view all the answers

    What is the unit of density in the given example?

    <p>lbm/ft3</p> Signup and view all the answers

    Why do we need to divide the mass by 32.2 in the American Engineering System?

    <p>To compensate for the unit of measurement</p> Signup and view all the answers

    What is the apparent weight of an object submerged in a fluid equal to?

    <p>Real weight minus buoyant force</p> Signup and view all the answers

    What is the specific gravity of an object defined as?

    <p>The weight of a given volume of the object divided by the weight of the same volume of water</p> Signup and view all the answers

    If an object weighs 300 lbs in air, and 200 lbs when submerged in water, what is the weight of the water displaced?

    <p>100 lbs</p> Signup and view all the answers

    What is Archimedes' Principle used for in the context of the given example?

    <p>To calculate the weight of the displaced fluid</p> Signup and view all the answers

    If an object has a specific gravity of 0.5, what is the weight of the same volume of water?

    <p>Twice the weight of the object</p> Signup and view all the answers

    What does the symbol xi represent in the given relationships?

    <p>The values of the random variable</p> Signup and view all the answers

    What is the formula to calculate the variance of the sample distribution?

    <p>$\sum\limits_{i=1}^{n} (xi - \bar{x})^2 / (n-1)$</p> Signup and view all the answers

    What is the value of the standard deviation for the given sample of x's?

    <p>1.966</p> Signup and view all the answers

    What does the area under the normal probability distribution curve between two values x1 and x2 represent?

    <p>The probability that a randomly selected value would fall between x1 and x2</p> Signup and view all the answers

    What is the symbol for the variance of the sample distribution?

    <p>σ^2</p> Signup and view all the answers

    What is the purpose of calculating the mean and standard deviation of a sample of x's?

    <p>To summarize the characteristics of the sample data</p> Signup and view all the answers

    What is the relationship between the standard deviation and the variance of the sample distribution?

    <p>The standard deviation is the square root of the variance</p> Signup and view all the answers

    What is the characteristic of the normal probability distribution that represents probability?

    <p>The area under the curve</p> Signup and view all the answers

    What is the sample space for the experiment of tossing a coin?

    <p>{head, tail}</p> Signup and view all the answers

    How many elements are in the sample space for the experiment of rolling a die?

    <p>6</p> Signup and view all the answers

    What is the sample space for the experiment of drawing a card from a deck of 52 cards?

    <p>A set of 52 possible outcomes</p> Signup and view all the answers

    What is the sample space for the government agency's experiment of deciding where to locate three new nuclear research laboratories?

    <p>{0, 1, 2, 3}</p> Signup and view all the answers

    If sets M1, M2, and M3 contain, respectively, N1, N2, and N3 elements, how many ways are there to select first an element from M1, then an element from M2, and finally an element from M3?

    <p>N1 × N2 × N3</p> Signup and view all the answers

    What is the characteristic of mutually exclusive events?

    <p>The events cannot occur at the same time.</p> Signup and view all the answers

    If an oil company has 15 candidates for the President and Vice President, how many ways are there to select the President and Vice President?

    <p>15 × 15</p> Signup and view all the answers

    What is the rule for finding the probability of either of two mutually exclusive events occurring?

    <p>Add the probabilities of the two events.</p> Signup and view all the answers

    What is the purpose of a sample space in an experiment?

    <p>To list all possible outcomes of the experiment</p> Signup and view all the answers

    What is the probability of getting an odd number in one roll of a die?

    <p>1/2</p> Signup and view all the answers

    What is the characteristic of a sample space?

    <p>It is a set of all possible outcomes</p> Signup and view all the answers

    If the probability of event A is 0.4 and the probability of event B is 0.3, what is the probability of either event A or event B occurring?

    <p>0.5</p> Signup and view all the answers

    What is the definition of independent events?

    <p>Events whose occurrence or non-occurrence has no effect on the outcome of the other event.</p> Signup and view all the answers

    What is the probability of drawing a king of spades or a 4 of diamonds from a 52-card deck?

    <p>2/52</p> Signup and view all the answers

    If event A and event B are mutually exclusive, what is the probability of both events occurring?

    <p>Zero.</p> Signup and view all the answers

    What is the probability of getting 1 or 6 in one roll of a die?

    <p>1/3 or 0.33</p> Signup and view all the answers

    What is the mass of CO2 in pounds, given the volume and density of solid CO2?

    <p>95 lbs</p> Signup and view all the answers

    What is the molecular weight of CO2?

    <p>44</p> Signup and view all the answers

    What is the value of R (gas constant) in ft³ × psia / lb mole × °K?

    <p>19.31</p> Signup and view all the answers

    What is the volume of the container in ft³, given the pressure and temperature of the gas?

    <p>744 ft³</p> Signup and view all the answers

    What is the pressure of the gas in psia, given the gauge pressure of 1000 psig?

    <p>1014.7 psia</p> Signup and view all the answers

    What is the temperature of the gas in °R, given the temperature of 80°F?

    <p>540 °R</p> Signup and view all the answers

    What is the number of moles of CO2, given the mass of CO2 in pounds?

    <p>2.16 lb moles</p> Signup and view all the answers

    What is the mass of hydrogen in pounds, given the volume and pressure of the container?

    <p>15 lbs</p> Signup and view all the answers

    What is the mole fraction of NaCl in a solution containing 100 moles of water and 4 moles of NaCl?

    <p>0.0384</p> Signup and view all the answers

    A 10 percent by weight brine solution contains 2000 pounds of salt. What is the weight of salt in the solution?

    <p>200 pounds</p> Signup and view all the answers

    What is the weight of sulfuric acid in 100 cubic feet of a 10 percent by volume sulfuric acid solution, given that the density of sulfuric acid is 70 pounds per cubic foot?

    <p>700 pounds</p> Signup and view all the answers

    How many pound moles of salt are in 2000 pounds of a 10 percent by weight brine solution, given that the molecular weight of salt is 58.5?

    <p>3.42</p> Signup and view all the answers

    What is the weight of 100 moles of water?

    <p>1800 grams</p> Signup and view all the answers

    A 10 percent by volume sulfuric acid solution contains 100 cubic feet of sulfuric acid. What is the total volume of the solution?

    <p>1000 cubic feet</p> Signup and view all the answers

    What is the molecular weight of NaCl?

    <p>58.5</p> Signup and view all the answers

    How many moles of water are in 1800 grams of water?

    <p>100</p> Signup and view all the answers

    What value of gas constant should be used if pressure is expressed in mm Hg, volume in ft3, moles in lb moles, and temperature in degrees Rankine?

    <p>28,300</p> Signup and view all the answers

    What is the unit of pressure in the formula R = 0.02366, when volume is in ft3, moles in gram moles, and temperature in degrees Rankine?

    <p>psia</p> Signup and view all the answers

    What is the value of R if pressure is expressed in in Hg, volume in liters, moles in lb moles, and temperature in degrees Kelvin?

    <p>37.2</p> Signup and view all the answers

    What is the unit of volume in the formula R = 0.02366, when pressure is in psia, moles in gram moles, and temperature in degrees Rankine?

    <p>ft3</p> Signup and view all the answers

    What value of gas constant should be used if pressure is expressed in atm, volume in liters, moles in gram moles, and temperature in degrees Kelvin?

    <p>0.08205</p> Signup and view all the answers

    What is the value of R if pressure is expressed in ft H2O, volume in ft3, moles in lb moles, and temperature in degrees Rankine?

    <p>44.6</p> Signup and view all the answers

    What is the unit of temperature in the formula R = 0.02366, when pressure is in psia, volume in ft3, and moles in gram moles?

    <p>degrees Rankine</p> Signup and view all the answers

    What is the value of R if pressure is expressed in psia, volume in liters, moles in lb moles, and temperature in degrees Rankine?

    <p>10.73</p> Signup and view all the answers

    What is the value of gc in the American Engineering System and British Engineering?

    <p>32.2</p> Signup and view all the answers

    What is the unit of mass in the British Engineering system of units?

    <p>Slug</p> Signup and view all the answers

    What is the unit of force in the Metric Absolute System (CGS)?

    <p>Dyne</p> Signup and view all the answers

    What is the value of gc in the Metric Absolute System (CGS) and SI Units?

    <p>1</p> Signup and view all the answers

    If an object with a mass of 2 kg accelerates at a rate of 1 m/s², what is the amount of force exerted on this object?

    <p>2 N</p> Signup and view all the answers

    What is the unit of length in the British Engineering System?

    <p>Foot</p> Signup and view all the answers

    What is the unit of time in the SI Units?

    <p>Second</p> Signup and view all the answers

    What is the unit of mass in the SI Units?

    <p>Kilogram</p> Signup and view all the answers

    What is the unit of velocity calculated from the equation V = C × RPM?

    <p>feet per minute</p> Signup and view all the answers

    What is the purpose of dividing the density by 32.2 in the American Engineering System?

    <p>To maintain consistency of units in the calculation</p> Signup and view all the answers

    What is the formula for calculating the hydrostatic pressure at the base of a fluid column?

    <p>P = Po + gh</p> Signup and view all the answers

    What is the unit of pressure in the given example?

    <p>lbf/ft2</p> Signup and view all the answers

    What is the value of the pressure at the bottom of the tank in the given example?

    <p>2116.8 lbf/ft2</p> Signup and view all the answers

    What is the purpose of converting the pressure from pounds per square inch to pounds per square foot?

    <p>To maintain consistency of units in the calculation</p> Signup and view all the answers

    What is the formula for calculating the velocity of the fluid in the given example?

    <p>V = C × RPM</p> Signup and view all the answers

    Why is the total pressure at the bottom of the tank calculated to be 2116.8 lbf/ft2?

    <p>Because the barometric pressure is 14.7 psi</p> Signup and view all the answers

    What is the formula for calculating the average speed of an object?

    <p>V = (s final - s initial) / (t final - t initial)</p> Signup and view all the answers

    What is the equation of motion for an object with constant acceleration?

    <p>s = V0t + at^2</p> Signup and view all the answers

    What is the unit of acceleration?

    <p>feet per second squared</p> Signup and view all the answers

    What is the formula for calculating the distance traveled by an object with constant acceleration?

    <p>s = V0t + at^2/2</p> Signup and view all the answers

    What is the initial speed of the object in the example problem?

    <p>5 ft/sec</p> Signup and view all the answers

    What is the acceleration of the object in the example problem?

    <p>2 ft/sec^2</p> Signup and view all the answers

    How far does the object travel after 10 seconds?

    <p>200 ft</p> Signup and view all the answers

    What is the purpose of writing down all the known and unknown values in solving problems?

    <p>To identify the equation to use to solve the problem</p> Signup and view all the answers

    What is the unit of pressure measured by a Bourdon gauge?

    <p>psi</p> Signup and view all the answers

    What is the purpose of a manometer?

    <p>To measure fluid pressure</p> Signup and view all the answers

    What is the formula to calculate the volume of a cylinder?

    <p>V = πD²H</p> Signup and view all the answers

    What is the density of water in the given example?

    <p>62.4 lbm/ft³</p> Signup and view all the answers

    What is the formula to calculate the buoyant force exerted on an object?

    <p>F = ρVg</p> Signup and view all the answers

    What is the unit of measurement for the buoyant force in the given example?

    <p>lbf</p> Signup and view all the answers

    What is the purpose of Archimedes' Principle?

    <p>To calculate the buoyant force</p> Signup and view all the answers

    What is the shape of a Bourdon gauge?

    <p>C-shaped</p> Signup and view all the answers

    What is the number of atoms in one gram mole of any element?

    <p>6.024 × 10²³</p> Signup and view all the answers

    What is the weight in grams of one gram mole of an element with an atomic weight of 32?

    <p>32 grams</p> Signup and view all the answers

    What is the number of gram moles of an element with an atomic weight of 23 in 46 grams of the element?

    <p>1 gram mole</p> Signup and view all the answers

    What is the relationship between the number of atoms in one gram mole of an element and Avogadro's number?

    <p>The number of atoms is equal to Avogadro's number.</p> Signup and view all the answers

    If one gram mole of an element contains 6.024 × 10²³ atoms, how many atoms are in 2 gram moles of the element?

    <p>12.048 × 10²³ atoms</p> Signup and view all the answers

    What is the weight in grams of one atom of an element with an atomic weight of 32?

    <p>0.016 grams</p> Signup and view all the answers

    If one gram mole of an element contains 6.024 × 10²³ atoms, how many gram moles are in 12.048 × 10²³ atoms of the element?

    <p>2 gram moles</p> Signup and view all the answers

    What is the relationship between the atomic weight of an element and the weight in grams of one gram mole of the element?

    <p>The atomic weight is equal to the weight in grams.</p> Signup and view all the answers

    What is the probability of getting a head when flipping a coin?

    <p>0.5 or 50 percent</p> Signup and view all the answers

    What is the probability of drawing a 4 out of a 52-card deck?

    <p>0.077 or 7.7 percent</p> Signup and view all the answers

    What is the probability of getting an even number in one roll of a die?

    <p>0.5 or 50 percent</p> Signup and view all the answers

    What is the probability of drawing a king or an ace from a deck of cards?

    <p>0.154 or 15.4 percent</p> Signup and view all the answers

    What is the characteristic of mutually exclusive events?

    <p>When one event occurs, the other event cannot occur</p> Signup and view all the answers

    What is the formula to calculate the probability of an event?

    <p>Number of favorable outcomes / Sample space</p> Signup and view all the answers

    What is the probability of getting an odd number in one roll of a die?

    <p>0.5 or 50 percent</p> Signup and view all the answers

    What is the purpose of a sample space in an experiment?

    <p>To list all possible outcomes of an experiment</p> Signup and view all the answers

    What is the probability of getting 1 or 6 in one roll of a die?

    <p>1/3</p> Signup and view all the answers

    What is the characteristic of mutually exclusive events?

    <p>If one event occurs, the other event cannot occur</p> Signup and view all the answers

    What is the rule for finding the probability of either of two mutually exclusive events occurring?

    <p>Add the probabilities of the two events</p> Signup and view all the answers

    What is the probability of drawing an ace of hearts or a king of spades or a 4 of diamonds from a 52-card deck?

    <p>3/52</p> Signup and view all the answers

    What is the definition of independent events?

    <p>Events that have no effect on the outcome of the other event</p> Signup and view all the answers

    What is the probability of getting an odd number in one roll of a die?

    <p>1/2</p> Signup and view all the answers

    If the probability of event A is 0.4 and the probability of event B is 0.3, what is the probability of either event A or event B occurring?

    <p>0.7</p> Signup and view all the answers

    What is the probability of voters who are either in favor of the legislation or undecided?

    <p>0.60</p> Signup and view all the answers

    What is the purpose of calculating the mean and standard deviation of a sample of x's?

    <p>To understand the characteristics of the population from which the sample is drawn</p> Signup and view all the answers

    What does the area under the normal probability distribution curve between two values x1 and x2 represent?

    <p>The probability that a randomly selected variable would assume a value between x1 and x2</p> Signup and view all the answers

    What is the characteristic of the normal probability distribution that represents probability?

    <p>The area under the curve</p> Signup and view all the answers

    What is the relationship between the standard deviation and the variance of the sample distribution?

    <p>The standard deviation is the square root of the variance</p> Signup and view all the answers

    What is the symbol for the variance of the sample distribution?

    <p>σ²</p> Signup and view all the answers

    What is the formula to calculate the variance of the sample distribution?

    <p>σ² = (Σ(x - μ)²) / (n - 1)</p> Signup and view all the answers

    What is the purpose of calculating Z1 and Z2?

    <p>To find the probability that a randomly selected variable would assume a value between x1 and x2</p> Signup and view all the answers

    What is the step to obtain the area between Z = 0, Z1 and Z2 from Table 3?

    <p>Obtain the area between Z = 0, Z1 and Z2 from Table 3</p> Signup and view all the answers

    What is the characteristic of mutually exclusive events?

    <p>They cannot occur together</p> Signup and view all the answers

    What is the rule for finding the probability of either of two mutually exclusive events occurring?

    <p>Add the probabilities of the two events</p> Signup and view all the answers

    What is the purpose of a sample space in an experiment?

    <p>To list all possible outcomes</p> Signup and view all the answers

    What is the probability of getting an odd number in one roll of a die?

    <p>1/2</p> Signup and view all the answers

    What is the characteristic of independent events?

    <p>The occurrence of one does not affect the probability of the other</p> Signup and view all the answers

    What is the probability of drawing a king of spades or a 4 of diamonds from a 52-card deck?

    <p>2/52</p> Signup and view all the answers

    If event A and event B are mutually exclusive, what is the probability of both events occurring?

    <p>0</p> Signup and view all the answers

    What is the probability of getting 1 or 6 in one roll of a die?

    <p>1/2</p> Signup and view all the answers

    What does the area under the normal probability distribution curve between two values x1 and x2 represent?

    <p>The probability that a randomly selected variable falls between x1 and x2</p> Signup and view all the answers

    To find the probability that a randomly selected variable assumes a value less than a given value x1, what do you do?

    <p>Add 0.5 to the area obtained from the table</p> Signup and view all the answers

    What is the purpose of calculating the mean and standard deviation of a sample of x's?

    <p>To summarize and describe the data</p> Signup and view all the answers

    What is the characteristic of the normal probability distribution that represents probability?

    <p>The curve is symmetric around the mean</p> Signup and view all the answers

    What is the formula to calculate the probability of a randomly selected variable assuming a value larger or smaller than a given value x1?

    <p>Z1 = (x1 - μ) / σ</p> Signup and view all the answers

    What is the probability that a randomly selected variable assumes a value less than 250 in a population with a mean of 200 and a standard deviation of 30?

    <p>95.15%</p> Signup and view all the answers

    What is the step to follow when dealing with a negative value of Z1?

    <p>Subtract the area obtained from the table from 0.5</p> Signup and view all the answers

    What does the probability of a randomly selected variable falling between two values x1 and x2 represent?

    <p>The area under the normal probability distribution curve between x1 and x2</p> Signup and view all the answers

    What is the characteristic of mutually exclusive events?

    <p>They can occur simultaneously</p> Signup and view all the answers

    If the probability of event A is 0.4 and the probability of event B is 0.3, what is the probability of either event A or event B occurring?

    <p>0.7</p> Signup and view all the answers

    What is the purpose of a sample space in an experiment?

    <p>To list all possible outcomes of an experiment</p> Signup and view all the answers

    What is the probability of getting an odd number in one roll of a die?

    <p>1/2</p> Signup and view all the answers

    If event A and event B are independent, what is the probability of both events occurring?

    <p>The product of their individual probabilities</p> Signup and view all the answers

    What is the probability of drawing a king of spades or a 4 of diamonds from a 52-card deck?

    <p>2/52</p> Signup and view all the answers

    If event A and event B are mutually exclusive, what is the probability of both events occurring?

    <p>0</p> Signup and view all the answers

    What is the definition of independent events?

    <p>Events whose occurrence does not affect each other</p> Signup and view all the answers

    What is the purpose of calculating the area under the normal probability distribution curve between two values Z1 and Z2?

    <p>To find the probability that a randomly selected variable would assume a value between x1 and x2</p> Signup and view all the answers

    What is the correct procedure when Z1 and Z2 have opposite signs?

    <p>Add the areas obtained for Z1 and Z2</p> Signup and view all the answers

    What is the characteristic of mutually exclusive events?

    <p>They cannot occur simultaneously</p> Signup and view all the answers

    What is the purpose of a sample space in an experiment?

    <p>To list all possible outcomes of an experiment</p> Signup and view all the answers

    What is the probability of getting an odd number in one roll of a die?

    <p>0.5</p> Signup and view all the answers

    What is the definition of independent events?

    <p>Events that have no effect on each other</p> Signup and view all the answers

    What is the rule for finding the probability of either of two mutually exclusive events occurring?

    <p>Add the probabilities of the two events</p> Signup and view all the answers

    What is the characteristic of a sample space?

    <p>It is a set of all possible outcomes of an experiment</p> Signup and view all the answers

    What is the probability that a randomly selected variable from a population with a mean of 200 and a standard deviation of 30 assumes a value less than 250?

    <p>95.15 per cent</p> Signup and view all the answers

    Given a population with a mean of 50 and a standard deviation of 10, what is the first step to find the probability that a randomly selected variable has a value greater than 65?

    <p>Calculate the value of Z1 from x1 - μ / σ</p> Signup and view all the answers

    What is the purpose of calculating the mean and standard deviation of a sample of x's?

    <p>To understand the characteristics of the normal probability distribution</p> Signup and view all the answers

    A population has a mean of 500 and a standard deviation of 50. What is the probability that a randomly selected variable from this population has a value larger than 600?

    <p>5.85 per cent</p> Signup and view all the answers

    What is the characteristic of the normal probability distribution that represents probability?

    <p>Area under the curve</p> Signup and view all the answers

    Given a population with a mean of 100 and a standard deviation of 20, what is the area under the curve between Z1 and Z2 if Z1 is 1.5 and Z2 is 2.5?

    <p>0.1567 - 0.0668</p> Signup and view all the answers

    What is the purpose of a sample space in an experiment?

    <p>To define the possible outcomes of an experiment</p> Signup and view all the answers

    What is the probability of either event A or event B occurring if the probability of event A is 0.4 and the probability of event B is 0.3?

    <p>0.4 + 0.3</p> Signup and view all the answers

    A population has a mean of 400 and a standard deviation of 40. What is the probability that a randomly selected variable from this population has a value larger than 450?

    <p>0.6772</p> Signup and view all the answers

    What is the probability that a randomly selected variable from a population with a mean of 300 and a standard deviation of 30 falls between 260 and 280?

    <p>0.0919</p> Signup and view all the answers

    A population has a mean of 600 and a standard deviation of 60. What is the probability that a randomly selected variable from this population assumes a value less than 580?

    <p>0.1587</p> Signup and view all the answers

    What is the probability that a randomly selected variable from a population with a mean of 200 and a standard deviation of 20 falls between 180 and 220?

    <p>0.4772</p> Signup and view all the answers

    A population has a mean of 800 and a standard deviation of 80. What is the probability that a randomly selected variable from this population has a value larger than 840?

    <p>0.1587</p> Signup and view all the answers

    What is the probability that a randomly selected variable from a population with a mean of 100 and a standard deviation of 10 falls between 90 and 110?

    <p>0.3413</p> Signup and view all the answers

    A population has a mean of 500 and a standard deviation of 50. What is the probability that a randomly selected variable from this population assumes a value less than 480?

    <p>0.1587</p> Signup and view all the answers

    What is the probability that a randomly selected variable from a population with a mean of 400 and a standard deviation of 40 falls between 380 and 420?

    <p>0.3413</p> Signup and view all the answers

    What is the probability of drawing a king of hearts or a 5 of diamonds from a 52-card deck?

    <p>1/13</p> Signup and view all the answers

    A coin is tossed twice. What is the probability of getting heads both times?

    <p>1/4</p> Signup and view all the answers

    What is the sample space for the experiment of rolling a die and tossing a coin?

    <p>36</p> Signup and view all the answers

    What is the probability of getting an odd number in one roll of a die, given that the number is greater than 3?

    <p>1/2</p> Signup and view all the answers

    What is the probability of getting either a 2 or a 5 in one roll of a die?

    <p>2/6</p> Signup and view all the answers

    If event A and event B are independent, what is the probability of both events occurring?

    <p>P(A) × P(B)</p> Signup and view all the answers

    What is the probability of getting exactly two heads in three tosses of a coin?

    <p>3/8</p> Signup and view all the answers

    A bag contains 5 red balls, 3 blue balls, and 2 green balls. What is the probability of drawing a blue ball?

    <p>3/10</p> Signup and view all the answers

    What does the subtraction of the area obtained for Z1 from that obtained for Z2 represent when both Z1 and Z2 are positive or both negative?

    <p>The probability of a randomly selected variable assuming a value between x1 and x2.</p> Signup and view all the answers

    What is the reason for adding the areas obtained for Z1 and Z2 when they have opposite signs?

    <p>Due to symmetry, the area for a negative value of Z1 is the same as the positive value of Z1 except that a negative Z1 has its area to the left of Z = 0.</p> Signup and view all the answers

    What is the purpose of using Table 3 in the problem of calculating the probability of a randomly selected variable assuming a value between x1 and x2?

    <p>To obtain the area between Z = 0 and a given value of Z1 or Z2.</p> Signup and view all the answers

    What is the formula for calculating Z1 and Z2 in the problem of calculating the probability of a randomly selected variable assuming a value between x1 and x2?

    <p>Z1 = x1 - μ / σ, Z2 = x2 - μ / σ</p> Signup and view all the answers

    What is the characteristic of the normal probability distribution that represents probability?

    <p>The area under the curve between two values.</p> Signup and view all the answers

    What is the purpose of calculating the probability of a randomly selected variable assuming a value between x1 and x2?

    <p>To determine the probability of a randomly selected variable assuming a value between x1 and x2.</p> Signup and view all the answers

    What is the relationship between the area under the normal probability distribution curve and the probability of a randomly selected variable assuming a value between x1 and x2?

    <p>The area under the curve is directly proportional to the probability.</p> Signup and view all the answers

    What is the formula for calculating the probability of a randomly selected variable assuming a value between x1 and x2, given the areas between Z = 0 and Z1, and Z = 0 and Z2?

    <p>P(x1 &lt; x &lt; x2) = Area between Z = 0 and Z1 + Area between Z = 0 and Z2.</p> Signup and view all the answers

    What is the primary factor that determines the type of bond formed between two atoms?

    <p>Electronegativity difference</p> Signup and view all the answers

    Which of the following reactions involves the gain of electrons by an atom?

    <p>Reduction</p> Signup and view all the answers

    What is the term for substances that have the same number of atoms of each element per molecule but differ in spatial arrangement?

    <p>Isomers</p> Signup and view all the answers

    What is the term for the energy associated with the formation or breaking of chemical bonds?

    <p>Heat of reaction</p> Signup and view all the answers

    What is the term for the substance in which the solute is dissolved?

    <p>Solvent</p> Signup and view all the answers

    What is the term for the amount of energy released or absorbed during a chemical reaction?

    <p>Heat of reaction</p> Signup and view all the answers

    What is the term for a substance that consists of an ordered array of its component atoms, ions, or molecules?

    <p>Crystalline substance</p> Signup and view all the answers

    Which of the following is an example of an oxidation-reduction reaction?

    <p>All of the above</p> Signup and view all the answers

    What is the relationship between the pressure, volume, and temperature of an ideal gas when its state is changed from conditions 1 to 2?

    <p>P1V1T2 = P2V2T1</p> Signup and view all the answers

    An ideal gas is contained in a 50 ft³ container at a temperature of 500 °R and a pressure of 30 psia. If the container is expanded to 100 ft³ and the temperature is increased to 600 °R, what is the new pressure of the gas?

    <p>20 psia</p> Signup and view all the answers

    What is the weight of one mole of sodium in grams?

    <p>23 grams</p> Signup and view all the answers

    What is the absolute temperature in °R of a gas at a temperature of 90 °F?

    <p>530 °R</p> Signup and view all the answers

    If one drop of seawater contains 300 billion atoms of sodium, how many atoms of sodium are in 2 drops of seawater?

    <p>600 billion atoms</p> Signup and view all the answers

    What is the mole fraction of oxygen in the given 1000 ft3 of air?

    <p>0.21</p> Signup and view all the answers

    An ideal gas is contained in a container at a pressure of 40 psia and a temperature of 400 °R. If the pressure is increased to 50 psia and the temperature is increased to 500 °R, what is the ratio of the initial volume to the final volume?

    <p>1.25</p> Signup and view all the answers

    What is the specific heat of a substance in calories per gram per degree Celsius if it is 2 Btu/lb?

    <p>2 calories/gram°C</p> Signup and view all the answers

    If a car travels a distance of 1200 miles in 24 hours, what is the average speed of this car?

    <p>50 mph</p> Signup and view all the answers

    What is the unit of volume in the American Engineering System?

    <p>ft³</p> Signup and view all the answers

    How much heat in Btu is needed to raise the temperature of 5 lbs of a substance by 10 degrees Fahrenheit if its specific heat is 2 Btu/lb?

    <p>100 Btu</p> Signup and view all the answers

    An ideal gas is contained in a container at a temperature of 300 °R and a pressure of 20 psia. If the temperature is increased to 400 °R and the pressure is increased to 30 psia, what is the ratio of the final volume to the initial volume?

    <p>1.2</p> Signup and view all the answers

    What is the sum of partial volumes of all components in a gas mixture?

    <p>Total volume of the mixture</p> Signup and view all the answers

    What is the absolute pressure in psia of a gas at a gauge pressure of 50 psig?

    <p>64.7 psia</p> Signup and view all the answers

    What is the vapor pressure of a liquid at equilibrium?

    <p>The pressure exerted by the vapor when the liquid and vapor phases have reached equilibrium.</p> Signup and view all the answers

    If the mole fraction of nitrogen is 0.79, what is the partial volume of nitrogen in 1000 ft3 of air?

    <p>790 ft3</p> Signup and view all the answers

    How many drops of seawater are needed to have 23 kilograms of sodium contained in salt?

    <p>2.008 × 10^15 drops</p> Signup and view all the answers

    An ideal gas is contained in a container at a temperature of 200 °R and a pressure of 10 psia. If the temperature is increased to 300 °R and the volume is increased to 200 ft³, what is the new pressure of the gas?

    <p>15 psia</p> Signup and view all the answers

    What is the definition of linear kinematics?

    <p>The analysis of linear motion without regard to its origin and effects</p> Signup and view all the answers

    What is the average speed of an object that travels a distance of 1000 miles in 20 hours?

    <p>50 mph</p> Signup and view all the answers

    What is the number of atoms of sodium in 23 kilograms of sodium?

    <p>6.024 × 10^26 atoms</p> Signup and view all the answers

    What is the unit of heat in the given example?

    <p>Btu</p> Signup and view all the answers

    What is the sum of mole fraction of oxygen and nitrogen in a gas mixture?

    <p>1.0</p> Signup and view all the answers

    What is the formula for calculating the average speed of an object?

    <p>V = s / t</p> Signup and view all the answers

    What is the total number of possible outcomes in an experiment where a set of 5 distinct objects is selected from a set of 10 distinct objects?

    <p>30240</p> Signup and view all the answers

    A sample space has 12 elements. If an event occurs 4 times out of 12, what is the probability of the event occurring?

    <p>1/3</p> Signup and view all the answers

    What is the number of ways to select first an element from set M1 with 5 elements, then an element from set M2 with 3 elements, and finally an element from set M3 with 2 elements?

    <p>30</p> Signup and view all the answers

    What is the sample space for an experiment where a card is drawn from a standard deck of 52 cards and the suit is of interest?

    <p>{heart, diamond, club, spade}</p> Signup and view all the answers

    What is the total number of elements in the sample space for an experiment where a coin is tossed 3 times?

    <p>8</p> Signup and view all the answers

    What is the probability of getting an even number in one roll of a die?

    <p>1/2</p> Signup and view all the answers

    What is the total number of ways to select 3 distinct objects from a set of 10 distinct objects?

    <p>120</p> Signup and view all the answers

    What is the sample space for an experiment where a government agency decides to locate 2 new research facilities out of 5 possible locations?

    <p>{0, 1, 2}</p> Signup and view all the answers

    What is the correct formula to relate the initial and final states of a gas?

    <p>P1V1 = P2V2</p> Signup and view all the answers

    What is the unit of temperature used in the given example?

    <p>Rankine</p> Signup and view all the answers

    What is the mole fraction of a component in a mixture?

    <p>The number of moles of the component divided by the total number of moles</p> Signup and view all the answers

    What is the purpose of using absolute pressure and temperature in the combined gas law equation?

    <p>To ensure consistency in the selected units</p> Signup and view all the answers

    What is the correct formula to calculate the mole fraction of a component in a mixture?

    <p>yA = nA / nt</p> Signup and view all the answers

    What is the correct unit to express the pressure in the combined gas law equation?

    <p>psia</p> Signup and view all the answers

    What is the relationship between the initial and final states of a gas under isometric conditions?

    <p>P1T2 = P2T1</p> Signup and view all the answers

    What is the purpose of the example that reworks the problem using the Kelvin temperature scale?

    <p>To illustrate the importance of consistency in the selected units</p> Signup and view all the answers

    What is the main reason why a molecule of sodium chloride is electrically neutral?

    <p>The number of electrons lost by the sodium atom equals the number of electrons gained by the chlorine atom.</p> Signup and view all the answers

    What type of bond is formed between atoms when they share their valence electrons?

    <p>Covalent bond</p> Signup and view all the answers

    What is the main difference between ionic and covalent bonds?

    <p>Ionic bonds are stronger than covalent bonds.</p> Signup and view all the answers

    What is the purpose of atomic absorption as an analytical technique?

    <p>To identify the presence of certain elements in a sample.</p> Signup and view all the answers

    What is the result of unequal sharing of electrons during bonding?

    <p>The creation of a polar molecule.</p> Signup and view all the answers

    Which of the following is an example of a covalent bond?

    <p>Hydrogen and oxygen.</p> Signup and view all the answers

    What is the term for the property of atoms that causes unequal sharing of electrons during bonding?

    <p>Electronegativity.</p> Signup and view all the answers

    Which type of bond is typically formed between atoms with very different electronegativity?

    <p>Ionic bond.</p> Signup and view all the answers

    What is the equation used to calculate the distance traveled by an object under constant acceleration?

    <p>s = V0t + gt^2</p> Signup and view all the answers

    What is the acceleration due to gravity in the given examples?

    <p>32.2 ft/s^2</p> Signup and view all the answers

    What is the final speed of the wrench when it reaches the ground?

    <p>56.7 ft/s</p> Signup and view all the answers

    How long does it take for the wrench to reach the ground?

    <p>4 seconds</p> Signup and view all the answers

    What is the equation used to calculate the final speed of an object under constant acceleration?

    <p>V^2 = V0^2 + 2gs</p> Signup and view all the answers

    What is the initial velocity of the wrench?

    <p>0 ft/s</p> Signup and view all the answers

    What is the height of the tower in the first example?

    <p>64.4 ft</p> Signup and view all the answers

    What is the purpose of using multiple equations to solve a problem?

    <p>To check the consistency of the results</p> Signup and view all the answers

    What is the condition required for an ideal gas to undergo an isothermal change?

    <p>Constant temperature</p> Signup and view all the answers

    If an ideal gas undergoes a change from P1 to P2 and V1 to V2, what is the relationship between the initial and final states?

    <p>P1V1 = P2V2</p> Signup and view all the answers

    A sample of gas is at a pressure of 2000 psig and a temperature of 70°F. What is the absolute pressure of the gas?

    <p>2014.7 psia</p> Signup and view all the answers

    An ideal gas undergoes a change from a volume of 10 liters to a volume of 40 liters. If the initial temperature is 300°C, what is the final temperature of the gas if the pressure is constant?

    <p>423°C</p> Signup and view all the answers

    What is the condition required for an ideal gas to undergo an isobaric change?

    <p>Constant pressure</p> Signup and view all the answers

    A sample of gas is at a volume of 10 liters and a temperature of 300°C. If the pressure is constant, what is the volume of the gas at a temperature of 150°C?

    <p>7.38 liters</p> Signup and view all the answers

    What is the relationship between the initial and final volumes of an ideal gas undergoing an isometric change?

    <p>V1 = V2</p> Signup and view all the answers

    A vessel contains an ideal gas at a temperature of 70°F. If the gas is transferred to another vessel with a volume four times the original vessel, what is the relationship between the initial and final pressures?

    <p>P1 &lt; P2</p> Signup and view all the answers

    What is the unit of the gas constant R?

    <p>pressure × volume / moles × temperature</p> Signup and view all the answers

    What is the absolute temperature in Kelvin if the temperature is 20°C?

    <p>293</p> Signup and view all the answers

    What is the relationship between gauge pressure and absolute pressure?

    <p>Absolute pressure is equal to gauge pressure plus atmospheric pressure</p> Signup and view all the answers

    What is the volume occupied by 100 pounds of a mixture if its density is 63.45 pounds per cubic foot?

    <p>1.58 cubic feet</p> Signup and view all the answers

    What is the unit of the value of R in the ideal gas law equation?

    <p>pressure × volume / moles</p> Signup and view all the answers

    What is the absolute temperature in Rankine if the temperature is 80°F?

    <p>580</p> Signup and view all the answers

    What is the relationship between the gas constant R and the ideal gas law equation?

    <p>R is equal to PV / nT</p> Signup and view all the answers

    What is the purpose of using absolute temperature scales in solving gas law problems?

    <p>To ensure that the temperatures are in the same unit</p> Signup and view all the answers

    What is the weight of CH4 in the gas mixture?

    <p>1000 lbs</p> Signup and view all the answers

    What is the molecular weight of H2?

    <p>2</p> Signup and view all the answers

    What is the total number of moles in the gas mixture?

    <p>595.83 lb moles</p> Signup and view all the answers

    What is the mole fraction of C2H6 in the gas mixture?

    <p>0.475</p> Signup and view all the answers

    What is the formula for calculating the partial volume of a component in an ideal gas mixture?

    <p>VA = yA * Vt</p> Signup and view all the answers

    What is the partial pressure of CH4 in the gas mixture?

    <p>210 psig</p> Signup and view all the answers

    What is the mole fraction of oxygen in air?

    <p>0.21</p> Signup and view all the answers

    What is the definition of partial volume?

    <p>The volume that would be occupied by a component of a mixture alone</p> Signup and view all the answers

    When an ideal gas undergoes an isothermal change, what remains constant?

    <p>Temperature only</p> Signup and view all the answers

    What is the formula for an ideal gas undergoing an isothermal change?

    <p>P1V1 = P2V2</p> Signup and view all the answers

    When an ideal gas undergoes an isobaric change, what remains constant?

    <p>Pressure only</p> Signup and view all the answers

    What is the formula for an ideal gas undergoing an isobaric change?

    <p>V1T2 = V2T1</p> Signup and view all the answers

    When an ideal gas undergoes an isometric change, what remains constant?

    <p>Volume only</p> Signup and view all the answers

    What is the condition for an ideal gas to undergo an isothermal change?

    <p>Constant temperature only</p> Signup and view all the answers

    What is the condition for an ideal gas to undergo an isobaric change?

    <p>Constant pressure only</p> Signup and view all the answers

    What is the number of atoms in one gram mole of any element?

    <p>6.024 × 10^23</p> Signup and view all the answers

    What is the condition for an ideal gas to undergo an isometric change?

    <p>Constant volume only</p> Signup and view all the answers

    What is the total number of moles in the given mixture of hydrogen and nitrogen?

    <p>20</p> Signup and view all the answers

    If the atomic weight of an element is 40, how many grams are in one gram mole of the element?

    <p>40 grams</p> Signup and view all the answers

    What is the number of atoms of sulfur in 32 grams of sulfur?

    <p>3.012 × 10^23 atoms</p> Signup and view all the answers

    What is the mole fraction of a component in a mixture?

    <p>The total number of moles of the component divided by the total number of moles</p> Signup and view all the answers

    What is the partial pressure of a component in an ideal gas mixture?

    <p>The mole fraction of the component multiplied by the total pressure</p> Signup and view all the answers

    If the weight of one atom of sodium is x grams, what is the weight of one gram mole of sodium?

    <p>23 grams</p> Signup and view all the answers

    A gas mixture contains 10 percent by weight methane, 5 percent hydrogen, and the balance ethane. What is the mole fraction of methane?

    <p>0.1</p> Signup and view all the answers

    What is the number of gram moles of sulfur in 128 kilograms of sulfur?

    <p>2,000 gram moles</p> Signup and view all the answers

    If the weight of one atom of an element is x grams, what is the weight of one gram mole of the element?

    <p>6.024 × 10^23 grams</p> Signup and view all the answers

    What is the condition for the sum of mole fractions of all components in a mixture?

    <p>The sum is equal to 1</p> Signup and view all the answers

    What is the mole fraction of nitrogen in the given mixture of hydrogen and nitrogen?

    <p>0.75</p> Signup and view all the answers

    What is the number of atoms of oxygen in 32 grams of oxygen?

    <p>3.012 × 10^23 atoms</p> Signup and view all the answers

    If the atomic weight of an element is 20, how many atoms are in 40 grams of the element?

    <p>1.2048 × 10^24 atoms</p> Signup and view all the answers

    What is the purpose of calculating the partial pressure of a component in a gas mixture?

    <p>To determine the properties of the individual components</p> Signup and view all the answers

    What is the unit of partial pressure?

    <p>psia</p> Signup and view all the answers

    What is the acceleration of gravity in ft/sec²?

    <p>32.2</p> Signup and view all the answers

    What is the equation for the instantaneous speed of an object under constant acceleration?

    <p>V = V0 + gt</p> Signup and view all the answers

    What is the equation for the distance covered by an object under constant acceleration?

    <p>s = V0t + gt²</p> Signup and view all the answers

    What is the initial speed of an object that is dropped from rest?

    <p>0</p> Signup and view all the answers

    What is the final speed of the object that takes 2 seconds to reach the ground?

    <p>64.4</p> Signup and view all the answers

    What is the purpose of the equation V² = V0² + 2gs?

    <p>To find the distance covered by an object</p> Signup and view all the answers

    What is the units of the acceleration of gravity?

    <p>ft/sec²</p> Signup and view all the answers

    What is the key concept in the problem of the object dropped from the top of a tower?

    <p>Constant acceleration</p> Signup and view all the answers

    Study Notes

    Avogadro's Number

    • Avogadro's number is 6.024 × 10^23, which is the number of atoms in one gram mole of any element.
    • One gram mole of an element is equal to the atomic weight of the element in grams.
    • Example: Oxygen has an atomic weight of 16, so one gram mole of oxygen is 16 grams, and it contains 6.024 × 10^23 atoms of oxygen.

    Calculating the Number of Atoms

    • To calculate the number of atoms in a given amount of an element, first calculate the number of gram moles, then multiply by Avogadro's number.
    • Example: 64 kilogram of sulfur has an atomic weight of 32, so it contains 2,000 gram moles, and therefore 12.048 × 10^26 atoms of sulfur.

    Weight of an Atom

    • The weight of an atom can be calculated using the formula W = mg, where W is the weight, m is the mass, and g is the acceleration of gravity.
    • Example: The weight of an atom of sodium is approximately 100 lbs.

    Apparent Weight

    • The apparent weight of an object submerged in a fluid is equal to its real weight minus the buoyant force exerted on the object by the displaced fluid.
    • Example: If the real weight of an object is 250 lbs, its apparent weight when submerged in water is 150 lbs.

    Specific Gravity

    • Specific gravity is defined as the weight of a given volume of an object divided by the weight of the same volume of water.
    • Example: If an object weighs 400 lbs and has an apparent weight of 300 lbs when submerged in a liquid, its specific gravity is 1.33.

    Standard Deviation and Variance

    • The standard deviation (σ) is a measure of the spread of a sample distribution.
    • The formula for standard deviation is σ = √(Σ(x_i - x̄)^2 / (n - 1)), where x_i is each value, x̄ is the mean, and n is the sample size.
    • Example: A sample of values has a mean of 4.33 and a standard deviation of 1.966.

    Normal Probability Distribution

    • The area under the normal probability distribution curve between two values represents the probability that a randomly selected value would fall between those values.
    • The normal probability distribution curve is symmetrical and bell-shaped.

    Set Theory, Probability, and Statistics

    • A sample space is a set of all possible outcomes of an experiment.
    • Example: The sample space for tossing a coin is {head, tail}, and the sample space for rolling a die is {1, 2, 3, 4, 5, 6}.
    • The sample space for drawing a card from a deck of 52 cards has 52 elements.

    Selecting Elements

    • If sets M1, M2, ..., Mk contain, respectively N1, N2, ..., Nk elements, there are N1 × N2 × ... × Nk ways of selecting first an element from M1, then an element from M2, ..., and finally an element from Mk.
    • Example: Selecting a President and Vice President from a list of 15 candidates for each position has 15 × 15 = 225 possible outcomes.

    Mutually Exclusive Events

    • Two events are mutually exclusive if they cannot occur simultaneously.
    • Example: Rolling an even number and rolling an odd number on a die are mutually exclusive events.
    • The probability of either A or B occurring is the sum of the probabilities of A and B.

    Independent Events

    • Two events are independent if the occurrence or non-occurrence of one event has no effect on the outcome of the other event.
    • The probability of two independent events occurring together is the product of their individual probabilities.

    Molecular Weight and Mole Fractions

    • Molecular weight of water is 18.
    • Molecular weight of NaCl is 58.5.
    • Moles of water can be calculated by dividing the given weight by the molecular weight of water.
    • Mole fraction of NaCl can be calculated by dividing the number of moles of NaCl by the sum of the number of moles of NaCl and water.

    Amount of Dissolved Solute

    • The amount of dissolved solute can be calculated using the formula: Amount of dissolved solute = Volume x Concentration in volume percent/100.
    • Alternatively, the amount of dissolved solute can be calculated using the formula: Amount of dissolved solute = Weight x Concentration in weight percent/100.

    Example Problems

    • In a 10% by volume sulfuric acid solution, the weight of sulfuric acid can be calculated by multiplying the volume of the solution by the density of sulfuric acid.
    • In a 10% by weight brine solution, the weight of salt can be calculated by multiplying the weight of the solution by the concentration of salt.

    Avogadro's Number

    • Avogadro's number is the number of atoms of an element in one gram mole of that element.
    • The values of Avogadro's number vary depending on the units of pressure, volume, moles, and temperature.

    Ideal Gas Law

    • The ideal gas law is given by the formula: PV = nRT.
    • The value of R (gas constant) depends on the units of pressure, volume, moles, and temperature.
    • Example: Calculate the volume occupied by the gas evolved from 1 ft3 of solid carbon dioxide.
    • Example: Calculate the mass of hydrogen in a 100 ft3 container under 1000 psig pressure at a temperature of 80°F.

    Systems of Units

    • There are different systems of units, including British Engineering, American Engineering, Metric Absolute, and SI Units.
    • The value of gc (gravitational constant) is equal to 1 in Metric Absolute and SI Units, but equal to 32.2 in British Engineering and American Engineering.

    Force and Mass

    • The second law of motion is given by the formula: F = ma.
    • The value of gc depends on the system of units used.
    • Example: Calculate the amount of force exerted on an object with a mass of 2 grams accelerating at a rate of 1 cm/sec2.

    Pressure

    • Pressure is the ratio of force to the area on which the force acts.
    • Pressure has units of force per area, such as newton/m2 (pascal) or lbf/in2 (psi).
    • Hydrostatic pressure at the base of a fluid column is given by the formula: P = Po + gh.
    • Example: Calculate the total pressure at the bottom of a tank containing water.

    Avogadro's Number

    • Avogadro's number is 6.024 × 10^23, which is the number of atoms in one gram mole of any element.
    • If we count the number of atoms in one gram mole of oxygen ( atomic weight 16), the result is 6.024 × 10^23 atoms of oxygen.

    Calculating the Number of Atoms

    • To calculate the number of atoms, we need to calculate how many gram moles of the element are contained in a given weight.
    • The formula to calculate gram moles is: gram mole = weight in grams / atomic or molecular weight.

    Example Problem 1

    • Calculate the number of atoms of sulfur contained in 64 kilograms of sulfur.
    • Solution: First, calculate the number of gram moles of sulfur (2000), then multiply by Avogadro's number to get the number of atoms (12.048 × 10^26).

    Example Problem 2

    • Calculate the weight in grams of one atom of sodium.
    • Solution: Since one gram mole of sodium (23 grams) contains 6.024 × 10^23 atoms, we can set up a proportion to find the weight of one atom.

    Kinematics

    • Constant acceleration: equations relating distance and speed to time are: V^2 = V0^2 + 2as, V = V0 + at, and s = V0t + at^2/2.
    • Example problem: An object has an initial speed of 5 ft/sec and an acceleration of 2 ft/sec^2. How far does it travel after 10 seconds?
    • Solution: Use the equation s = V0t + at^2/2 to find the distance traveled.

    Fluid Pressure Measurement

    • Bourdon gauge: a hollow tube closed at one end and bent into a C configuration, used to measure fluid pressure.
    • Manometer: a U-shaped tube partially filled with a liquid of known density, used to measure pressure difference.
    • Archimedes' Principle: the buoyant force on a submerged object is equal to the weight of fluid displaced by the object.

    Example Problem 3

    • A cylindrical object is completely submerged in water. What is the buoyant force exerted on the object?
    • Solution: Calculate the volume of the cylinder, then find the weight of the displaced water to get the buoyant force.

    Probability

    • Definition: probability is a number between 0 and 1 that represents the likelihood of an event occurring.
    • Example problem: What is the probability of getting a head in one flip of a coin?
    • Solution: The probability is 0.5 or 50%.

    Mutually Exclusive Events

    • Definition: two events are mutually exclusive if, when one event occurs, the other cannot, and vice versa.
    • Axiom: the probability of either A or B occurring is the SUM of the probabilities of A and B.
    • Example problem: If the proportion of voters favoring legislation is 0.38 and the proportion of voters who are undecided is 0.22, what is the proportion of voters who are either in favor of the legislation or undecided?
    • Solution: The probabilities are additive, so the answer is 0.38 + 0.22 = 0.60.

    Independent Events

    • Definition: two events are independent if the occurrence or non-occurrence of one event has no effect on the outcome of the other event.
    • Example problem: What is the probability of drawing an ace of hearts or a king of spades or a 4 of diamonds from a 52-deck of cards?
    • Solution: The events are mutually exclusive, so the probabilities are additive.

    Normal Probability Distribution

    • To find the probability that a randomly selected variable falls between two values, follow these steps:
    • Calculate Z1 and Z2 from (x1 - μ) / σ and (x2 - μ) / σ
    • Obtain the areas between Z = 0 and Z1, and Z = 0 and Z2 from Table 3
    • Subtract the area obtained for Z1 from that obtained for Z2 if both are positive or negative, or add the areas if they have opposite signs

    Example 2

    • A population is normally distributed with a mean of 50 and a standard deviation of 10
    • Find the probability that a randomly selected variable falls between 35 and 40
    • Calculate Z1 = (35 - 50) / 10 = -1.5 and Z2 = (40 - 50) / 10 = -1.0
    • Obtain areas from Table 3: 0.4332 for Z1 and 0.3413 for Z2
    • Calculate the probability: 0.4332 - 0.3413 = 0.0919 or 9.19%

    Finding Probability of a Randomly Selected Variable

    • To find the probability that a randomly selected variable assumes a value larger or smaller than a given value x1:
    • Calculate Z1 from (x1 - μ) / σ
    • If Z1 is positive:
      • For probability of a value greater than Z1, subtract the area obtained from 0.5
      • For probability of a value smaller than Z1, add 0.5 to the area obtained
    • If Z1 is negative:
      • For probability of a value greater than Z1, add 0.5 to the area obtained
      • For probability of a value smaller than Z1, subtract the area obtained from 0.5

    Examples 3 and 4

    • Example 3: Find the probability that a randomly selected variable from a population with mean 200 and standard deviation 30 assumes a value less than 250
    • Calculate Z1 = (250 - 200) / 30 = 1.66
    • Obtain area from Table 3: 0.4515
    • Calculate the probability: 0.5 + 0.4515 = 0.9515 or 95.15%
    • Example 4: Find the probability that a randomly selected variable from a population with mean 500 and standard deviation 50 has a value larger than 600
    • Calculate Z1 = (600 - 500) / 50 = 2.0
    • Obtain area from Table 3
    • Calculate the probability (not shown in the original text)

    Normal Probability Distribution

    • To find the probability that a randomly selected variable falls between two values, follow these steps:
    • Calculate Z1 and Z2 from (x1 - μ) / σ and (x2 - μ) / σ
    • Obtain the areas between Z = 0 and Z1, and Z = 0 and Z2 from Table 3
    • Subtract the area obtained for Z1 from that obtained for Z2 if both are positive or negative, or add the areas if they have opposite signs

    Example 2

    • A population is normally distributed with a mean of 50 and a standard deviation of 10
    • Find the probability that a randomly selected variable falls between 35 and 40
    • Calculate Z1 = (35 - 50) / 10 = -1.5 and Z2 = (40 - 50) / 10 = -1.0
    • Obtain areas from Table 3: 0.4332 for Z1 and 0.3413 for Z2
    • Calculate the probability: 0.4332 - 0.3413 = 0.0919 or 9.19%

    Finding Probability of a Randomly Selected Variable

    • To find the probability that a randomly selected variable assumes a value larger or smaller than a given value x1:
    • Calculate Z1 from (x1 - μ) / σ
    • If Z1 is positive:
      • For probability of a value greater than Z1, subtract the area obtained from 0.5
      • For probability of a value smaller than Z1, add 0.5 to the area obtained
    • If Z1 is negative:
      • For probability of a value greater than Z1, add 0.5 to the area obtained
      • For probability of a value smaller than Z1, subtract the area obtained from 0.5

    Examples 3 and 4

    • Example 3: Find the probability that a randomly selected variable from a population with mean 200 and standard deviation 30 assumes a value less than 250
    • Calculate Z1 = (250 - 200) / 30 = 1.66
    • Obtain area from Table 3: 0.4515
    • Calculate the probability: 0.5 + 0.4515 = 0.9515 or 95.15%
    • Example 4: Find the probability that a randomly selected variable from a population with mean 500 and standard deviation 50 has a value larger than 600
    • Calculate Z1 = (600 - 500) / 50 = 2.0
    • Obtain area from Table 3
    • Calculate the probability (not shown in the original text)

    Normal Probability Distribution

    • To find the probability that a randomly selected variable falls between two values, follow these steps:
    • Calculate Z1 and Z2 from (x1 - μ) / σ and (x2 - μ) / σ
    • Obtain the areas between Z = 0 and Z1, and Z = 0 and Z2 from Table 3
    • Subtract the area obtained for Z1 from that obtained for Z2 if both are positive or negative, or add the areas if they have opposite signs

    Example 2

    • A population is normally distributed with a mean of 50 and a standard deviation of 10
    • Find the probability that a randomly selected variable falls between 35 and 40
    • Calculate Z1 = (35 - 50) / 10 = -1.5 and Z2 = (40 - 50) / 10 = -1.0
    • Obtain areas from Table 3: 0.4332 for Z1 and 0.3413 for Z2
    • Calculate the probability: 0.4332 - 0.3413 = 0.0919 or 9.19%

    Finding Probability of a Randomly Selected Variable

    • To find the probability that a randomly selected variable assumes a value larger or smaller than a given value x1:
    • Calculate Z1 from (x1 - μ) / σ
    • If Z1 is positive:
      • For probability of a value greater than Z1, subtract the area obtained from 0.5
      • For probability of a value smaller than Z1, add 0.5 to the area obtained
    • If Z1 is negative:
      • For probability of a value greater than Z1, add 0.5 to the area obtained
      • For probability of a value smaller than Z1, subtract the area obtained from 0.5

    Examples 3 and 4

    • Example 3: Find the probability that a randomly selected variable from a population with mean 200 and standard deviation 30 assumes a value less than 250
    • Calculate Z1 = (250 - 200) / 30 = 1.66
    • Obtain area from Table 3: 0.4515
    • Calculate the probability: 0.5 + 0.4515 = 0.9515 or 95.15%
    • Example 4: Find the probability that a randomly selected variable from a population with mean 500 and standard deviation 50 has a value larger than 600
    • Calculate Z1 = (600 - 500) / 50 = 2.0
    • Obtain area from Table 3
    • Calculate the probability (not shown in the original text)

    Chemical Bonding

    • Large differences in electronegativity result in complete transfer of electrons from one atom to another, forming an ionic bond.
    • The sum of the energy of the chemical bonds in the products is usually different from the bond energies in the reactants, resulting in a heat of reaction.
    • Chemical reactions can produce or consume energy, categorized as endothermic (consumes energy) or exothermic (produces energy).
    • Isomers are substances with the same number of atoms of each element per molecule, but differ in spatial arrangement.
    • Crystals are ordered arrays of their component atoms, ions, or molecules.

    Oxidation-Reduction Reactions

    • Oxidation is the loss of electron(s) by an atom involved in a chemical reaction, accompanied by reduction.
    • Reduction is the gain of electron(s) by an atom involved in a chemical reaction, accompanied by oxidation.
    • Examples: Fe → Fe²⁺ + 2 electrons (oxidation), Cl₂ + 2 electrons → 2 Cl⁻ (reduction).

    Solutions

    • Solute is the dissolved substance.
    • Solvent is the substance in which the solute is dissolved.
    • Concentration of a solution can be expressed in physical units (e.g., weight of solute per unit volume of solution) or chemical units (e.g., weight percentage).
    • Example: 10 grams of NaCl per liter of solution, or 23 kg of sodium contained in salt.

    Vapor Pressure

    • Vapor pressure is the pressure exerted by the vapor when the liquid and vapor phases have reached equilibrium.
    • Vapor pressure increases with an increase in temperature.
    • Heat is energy in transfer.

    Specific Heat

    • Specific heat is the amount of heat required to raise the temperature of 1 gram of substance by one degree Celsius or one degree Fahrenheit.
    • Example: The specific heat of a substance is 2 Btu/lb, so how much heat is needed to raise the temperature of 5 lbs of this substance by 10 degrees Fahrenheit?

    Combined Gas Law

    • The relationship between pressure, temperature, and volume is given by: P₁V₁T₂ = P₂V₂T₁ or PV₁/T₁ = PV₂/T₂.
    • Example: An ideal gas is contained in a 100 ft³ expandable container, initially at 50 psig and 70°F. If the volume is doubled and the temperature is raised to 100°F, what is the new pressure?

    Linear Kinematics

    • Linear kinematics is the analysis of linear motion without regard to its origin and effects.
    • Average speed is equal to the distance covered in a certain time period divided by that amount of time.
    • Example: A car travels a distance of 950 miles in 18 hours. What is the average speed of this car?

    Set Theory, Probability, and Statistics

    • Sample space is the set of all possible outcomes of an experiment.
    • Example: The sample space for the experiment tossing a coin is {head, tail}, and for rolling a die is {1, 2, 3, 4, 5, 6}.
    • Example: The sample space for the experiment drawing a card from a deck of 52 cards is the set of all possible outcomes, which has 52 elements.
    • Selecting elements: If sets M₁, M₂, ..., Mᵏ contain, respectively, N₁, N₂, ..., Nᵏ elements, there are N₁ × N₂ × ... × Nᵏ ways of selecting first an element from M₁, then an element from M₂, ..., and finally an element from Mᵏ.

    Ionic Bonds

    • In a molecule of sodium chloride, the sodium ion is positively charged, and the chlorine ion is negatively charged.
    • However, sodium chloride as a compound is electrically neutral because the number of electrons lost by the sodium atom equals the number of electrons gained by the chlorine atom.

    Covalent Bonds

    • Covalent bonds occur when atoms share their valence electrons to complete the electron requirement in the valence shell of each atom.
    • Covalent bonds are weaker than ionic bonds.
    • An example of covalent bond formation is the reaction between hydrogen and oxygen to form water.

    Atomic Absorption

    • Atomic absorption is an analytical technique that takes advantage of the characteristic absorption of certain wavelengths of light by metals.

    Electronegativity

    • Electronegativity is a property associated with certain atoms that causes unequal sharing of electrons during bonding.
    • This results in shared electrons spending more time with one atom and less time with the other.

    Ideal Gas Law

    • PV = nRT
    • p is the absolute pressure (gauge pressure plus atmospheric pressure)
    • v is the volume occupied by the gas
    • n is the number of moles of gas
    • R is the universal gas constant whose value depends on the units selected for P, V, n, and T
    • T is the absolute temperature (degrees Fahrenheit plus 460 or degrees centigrade plus 273)

    Absolute Pressure and Temperature Scales

    • Absolute pressure = gauge pressure + atmospheric pressure
    • Two absolute temperature scales are used: degrees Kelvin (°C + 273) and degrees Rankine (°F + 460)

    Gas Constant

    • The value of the gas constant in the ideal gas law depends on the units chosen for pressure, temperature, number of moles, and volume.
    • Remember to use absolute pressure and temperature in solving gas law problems.

    Isothermal Changes in an Ideal Gas

    • When an ideal gas undergoes a change at constant temperature, T1 and T2 are equal.
    • P1V1 = P2V2

    Isobaric Changes in an Ideal Gas

    • When an ideal gas undergoes changes under constant pressure, the values of P1 and P2 are equal.
    • V1/V2 = T1/T2

    Isometric Changes in an Ideal Gas

    • When an ideal gas undergoes changes under constant volume conditions, the values of V1 and V2 are equal.
    • P1/P2 = T1/T2

    Mole Fraction

    • The number of moles of a component in a mixture divided by the total number of moles.
    • yA = nA/nt

    Examples and Practice Problems

    • Various examples and practice problems are provided to illustrate the application of gas laws, including problems involving isothermal, isobaric, and isometric changes.

    Avogadro's Number

    • Avogadro's number is 6.024 × 10^23, which means that if we count the number of atoms in one gram mole of any element, the result is 6.024 × 10^23 atoms.
    • Example: The atomic weight of oxygen is 16, and one gram mole of oxygen weighs 16 grams. Therefore, the number of atoms of oxygen in 16 grams of oxygen (one gram mole) is 6.024 × 10^23 atoms of oxygen.

    Calculating Number of Atoms

    • To calculate the number of atoms of an element, we need to know the atomic weight and the weight of the substance in grams.
    • Example: The atomic weight of sulfur is 32, and we want to calculate the number of atoms of sulfur in 64 kilogram of sulfur. We first calculate the number of gram moles of sulfur (2000) and then multiply it by Avogadro's number to get the number of atoms of sulfur (12.048 × 10^26 atoms).

    Calculating Weight of an Atom

    • To calculate the weight of an atom, we need to know the atomic weight and Avogadro's number.
    • Example: The atomic weight of sodium is 23, and we want to calculate the weight of one atom of sodium. We know that one gram mole of sodium (23 grams) contains 6.024 × 10^23 atoms of sodium. Therefore, the weight of one atom of sodium is 23 grams / 6.024 × 10^23 atoms.

    Gas Laws

    • Isothermal changes in an ideal gas: P1V1 = P2V2
    • Isobaric changes in an ideal gas: V1/T1 = V2/T2
    • Isometric changes in an ideal gas: V1 = V2

    Partial Pressure and Partial Volume

    • The partial pressure of a component in an ideal gas mixture is equal to its mole fraction multiplied by the total pressure: PA = yA Pt
    • The partial volume of a component of an ideal gas mixture equals its mole fraction in the mixture multiplied by the total mixture volume: VA = yA Vt

    Motion due to Constant Gravitational Acceleration

    • The equations for speed, time, and distance covered for falling objects due to constant acceleration of gravity are:
      • V = V0 + gt
      • s = V0t + (1/2)gt^2
      • V^2 = V0^2 + 2gs
    • Example: An object is dropped from the top of a tower and takes 2 seconds to reach the ground. What is the speed of the object just as it reaches the ground? What is the height of the tower?

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    Mathematics and Chemistry Calculations PDF

    Description

    Learn about Avogadro's number, its value, and its relationship with gram mole. Understand how to calculate the weight of an element using atomic weight and Avogadro's number.

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