Mathematical Concepts and Fundamentals of Chemistry Chapter 1 PDF

Review of Mathematical Concepts and Fundamentals of Chemistry CHAPTER 1 CHEMISTRY -STUDY OF THE COMPOSITION, STRUCTURE AND PROPERTIES OF MATTER AND THE CHANGES IT UNDERGOES. STATES OF MATTER SOLID LIQUID GAS  OTHER STATES OF MA...

Review of Mathematical Concepts and Fundamentals of Chemistry CHAPTER 1 CHEMISTRY -STUDY OF THE COMPOSITION, STRUCTURE AND PROPERTIES OF MATTER AND THE CHANGES IT UNDERGOES. STATES OF MATTER SOLID LIQUID GAS  OTHER STATES OF MATTER PLASMA -IS AN IONIZED GAS. BOSE-EINSTEIN CONDENSATE OCCURS AT ULTRA LOW TEMPERATURE(VERY NEAR TO ABSOLUTE ZERO, CLOSE TO THE POINT THAT THE ATOMS ARE NOT MOVING AT ALL). BOSONS – are particles that carry energy and forces throughout the universe. CLASSIFICATIONS OF MATTER Chemistry and Engineering “Why do I have to take chemistry anyway? I’ll never really need to know any of this to be an engineer.” Algebraic Manipulation in Chemistry Sample 1.1. Determine the following unknown quantity from the following equation a. Given: pH = - log [H+] Required: [H+] = ? mass b. Given:  = V Required: V=?  d3 c. Given: V= 6 Required: d=? Practice Exercise 1.1. Solve for the unknown quantity from the given formula y2 – y1 Cc Dd a. Given: m= x2 – x1 b. Given: Q = Required: x2 = ? Aa Bb Required: a = ?  𝒅𝟐 𝐡 ln k2– ln k1= 𝐸𝑎 𝑇2 −𝑇1 V=  ; h=? 𝑅𝑇1 𝑇2 ; Ea = ? 4 k2 =? VARIATIONS Many laws in Chemistry are stated as variation statements. These statements may be direct, inverse or joint variations. Charle’s Law V∝T Boyle’s Law V ∝ 1/P Or P ∝ 1/V Specific Heat Formula Q ∝ m∆T Direct Proportion Direct proportion is a mathematical comparison between two numbers where the ratio of the two numbers is equal to a constant value. The proportion definition says that when two ratios are equivalent, they are in proportion. The definition of direct proportion states that "When the relationship between two quantities is such that if we increase one, the other will also increase, and if we decrease one the other quantity will also decrease, then the two quantities are said to be in a direct proportion". Example: If there are two quantities x and y where x = number of candies and y = total money spent. If we buy more candies, we will have to pay more money, and we buy fewer candies then we will be paying less money. So, here we can say that x and y are directly proportional to each other. It is represented as x ∝ y. Direct proportion is also known as direct variation. Example 2: If the cost of 8 pounds of apples is $10, what will be the cost of 32 pounds of apples? Solution: It is given that, Weight of apples = 8 lb Cost of 8 lb apples = $10 Let us consider the weight by x parameter and cost by y parameter. To find the cost of 32 lb apples, we will use the direct proportion formula. y=kx 10 = k × 8 (on substituting the values) k = 5/4 Now putting the value of k = 5/4 when x = 32 we have, The cost of 32 lb apples = 5/4 × 32 y =5×8 y = 40 Answer: The cost of 32 lb apples is $40. What is Inverse Proportion? The definition of inverse proportion states that "Two quantities are said to be in inverse proportion if an increase in one leads to a decrease in the other quantity and a decrease in one leads to an increase in the other quantity". In other words, if the product of both the quantities, irrespective of a change in their values, is equal to a constant value, then they are said to be in inverse proportion. For example, let us take the number of workers and the number of days required by them to complete a given amount of work as x and y respectively. Numbers of Workers (x) Number of Days Required (y) 16 3 12 4 8 6 4 12 Each row, the product of x and y are the same. That means if there are 16 workers, they will complete the work in 3 days. So, x × y = 16 × 3 = 48. A decrease in number of workers, less number of workers will do the same work in more time. But the product of x and y here, it is 12 × 4 = 48. Again, for 8 workers in 6 days, the product is 48. And same for 4 workers in 12 days. So, the product of two quantities in inverse proportion is always equal. The inverse proportional formula is written as y = k/x where k is the constant of proportionality. y increases as x decreases. y decreases as x increases.  Suppose x and y are in an inverse proportion such that when x = 120, y = 5. Find the value of y when x = 150 using the inverse proportion formula. Solution: Given: x = 120 when y = 5. x ∝ 1/y x = k / y, where k is a constant, or k = xy To find: Value of y. Putting, x = 120 and y = 5, we get; k = 120 × 5 = 600 Now, when x = 150, then; 150 y = 600 y = 600/150 = 4 That means when x is increased to 150 then y decreases to 4. Answer: The value of y is 4 when x = 150. Example: a machine manufactures 20units per hour The units that machine manufactures is directly proportional to how many hours it has worked. More works the machine does, more are the units manufactured; in direct proportion. This could be written as: Units ∝ Hours Worked If it works 2 hours we get 40 Units If it works 4 hours we get 80 Units Sample 1. A quantity L varies directly as another quantity m. Write the equation and solve for m. Sample 2. A quantity y varies directly as x. When x is 6, y is 4. What should be the value of x for y to be 12? Sample 3. In traveling a given distance, the time travel varies inversely as the velocity of the vehicle. If it takes a car 3 hours to travel a certain distance at the rate of 60 kph, how long will it take the same car to cover the same distance if it travels at the rate of 90 kph? 1.2 Systems of Measurements Measurements is simply a way of comparing certain attribute of an object with some given standard. The International Standards of mass equal to 1 kg refer to a cylinder of platinum-iridium. The second is defined as 9, 192,631,770 times the period of vibration of radiation from Cesium atom. 1 meter is defined as the distance of light travels in a vacuum in 1/299,792,458 seconds. 1.2 Systems of Measurements Measurement may be taken by DIRECT METHOD or by INDIRECT METHOD. A direct measurement is a item that with ease can be measured such as the height of a toddler. An indirect measurement is the measurement of an item that is difficult to measurement. The measurement can be found using similar triangles, the Pythagorean Theorem and Thales Shadow Theorem. 1.2 Systems of Measurements Fundamental quantities are basic measurable quantities that have no connection with each other. Table 1.2 Seven SI Fundamental Quantities Quantity Symbol Unit Length l meter (m) Mass m kilogram (kg) Time t second (s) Electric I ampere (A) Current Temperature T Kelvin (K) Amount of n mole (mol) substance Luminous Iv candela (cd) Intensity  Derived quantities are quantities that are expressed as a combination of the fundamental quantities. Table 1.3 Some SI Derived Units Quantity Sy Unit Area mb A square meter, m2 Volume ol V cubic meter, m3 Density ρ kilogram per cubic meter, kg/m3 Electric charge Q coulomb, C (A-s) Electric potential Ɛ volt, V (J/C) Pressure P Pascal, Pa Energy E Joule, J Force F Newton, N (kg∙m/ s2) Velocity v meter per second (m/s) acceleration a meter per second square(m/s2) Work W Joule, J (kg∙m2/ s2) Power P Watt, W (J/s)  SYSTEM OF UNITS It is a complete set of units both fundamental and derived for all kind of quantities. Old British or English System FPS (foot-pound-second) Metric System SI (Systeme International d ‘Unites) MKS (meter-kilogram-second) CGS(centimeter-gram-second) 1.2 Systems of Measurements The International System of Units (SI) Table 1.1 Commonly Used Prefixes and Numerical Values for SI Units Practice: 1. Convert 4.0 cm to m. 2. Convert 4 nL to L. 3. Convert 40.0 L to μL 4. Convert 40 km to mm 5. Convert 0.04 GL to mL. Significant Figures Significant figures are those digits in a number with known certainty plus the first digit that is uncertain. Exact numbers (constant number) are numbers that have defined values and are obtained by definition. This number is not counted in determining the number of significant figures. For example, 12 eggs in a dozen, and 2 pieces in a pair. Another example, numbers are written in a formula is an exact number. Like in the formula for the (a) area of a solid rectangular box, A = 2( lt + lw + wt), 2 is considered exact number; (b) V = r2h,  is an exact number, and all others are called inexact numbers or variable numbers. Inexact numbers (variable numbers) are numbers obtained by measurement and must be expressed in the right significant figures. The number of significant figures must depend on the sensitivity of measuring devices used. Rule 1 All nonzero digits are significant. 82,824 88,000 80,000 88,800 Rule 2 Middle zeros (zeros between two nonzero digits) are significant. 3,091 109,003 82,024 100,930 100,903 Rule 3 All zeros that are written to the right of the decimal number count as significant figures. 73.0 1254.10 4,531.01 5607.400 83.000 Rule 4 Leading zeros (zeros to the right of the decimal number less than one and preceding a nonzero digit) are not significant. 0.000491 0.000006 0.031 0.1120 Rule 5 Trailing zeros (zeros at the end of a nonzero digit) is significant if a decimal point or a bar on top is indicated, for example, 300. kW has three significant figures; 10ô0 has three significant figures, and so on.  How many significant figures are shown in each of the following? a. 2900 m b. 0.000002 cm c. 1.0 x 105 Pa d. 450.0 mL e. 0.00530 mol f. 3008 g g. 2.355 mm h. 0.40 cm i. 0.125 in  Scientific Notation Scientific notation is a convenient way of expressing a very small or a very large number. 24,984,000,000,000 miles (distance of Earth to the closest star) 0.0000000203 seconds (amount of time of light to go from one side of an average size bedroom to other side ) 30,600,00,000,000,000,000,000,000 atoms (number of atoms in an average size of glass) 0.000 000 000 000 000 000 000 000 00369 lbs. (weight of proton in an atom) Scientific Notation Scientific notation is a convenient way of expressing a very small or a very large number. It simplifies the use of significant figures. It has the form M x 10n M =decimal number with one nonzero digit before the decimal point n= is a positive or negative whole number. Express in proper scientific notation: (a)0.002854 = (2.854 x 10-3) (b) 12,000,000 = (1.2 x 107)  Express the following numbers in scientific notation and express in 3 SF. a. 706.5 b. 0.00123 c. 125,000 d. 0.00000034 e. 3,000,000  Computational Rules Rule 1 In addition or subtraction, the number of decimal places to the decimal point's right in the final sum and difference is determined by the lowest number of decimal places to the decimal point's right in any of the original numbers. The following examples illustrate this rule. Sample 1. 4.1 0. 0521 + 1. 3456 3. 56 4. 9577 (4.96) Sample 1. 4.2 156. 23 - 4. 5 151. 73 (151.7) Rule In multiplication or division, the result must contain 2 the same number of significant figures as in the original number having the smallest number of significant figures. Consider the following examples: Sample 1.4.3 27.3 x 1.2 = 32.76 (33) Sample 1.4.4 70.05 x 31.5 = 147.105 (150) 15 a. Perform the indicated operation and express answers in the right significant figure. 2.5 m - 0.31m 3.583 s 1.43 x 5 396/23 7.0032 – 3.05 1.234 + 2.3 + 0.007 1.5 Conversion of Units Dimensional analysis, or factor unit method, is used to convert a unit to another unit, a numerator and a denominator representing a relationship are used. Table 1.4 Common Conversion Factors 1 m = 3.281 ft = 1.094 yards (yd) 1 inch (in) = 2.54 cm LENGTH 1 foot (ft) = 12 in 1 mile (mi) = 1.61 km 1 square meter (m2) = 10.76 square feet (ft2) 1 ft2= 929 square centimeter (cm2) AREA 1 hectare = 10,000 m2 = 2.471 acres 1 acre = 43,560 ft2 1 mL = 1 cubic centimeter (cc or cm3) 1 L = 1000 mL = 1.057 quarts (qt) VOLUME 1 cubic meter (m3) = 1000 L 1 gal = 3.7854 L 1 earth year = 365.25 days 1 week = 7 days TIME 1 day = 24 hours (hr) 1 hr = 60 minutes (min) = 3600 sec 1 min = 60 seconds (s) 1 kg = 1000 g = 2.2 pounds (lb) 1 g = 1000 mg MASS 1. metric ton = 1000 kg 2. pound = 454 g Sample 1.5.1: The height of a building is 100.0 feet. Find the height in meters in the right significant figures 1m h = 100.0 ft x = 30.48 m 3.281 ft Sample 1.5.2 : Convert 125 mm2 to cm2 (1 cm)2 A = 125 mm2 x (10mm)2 = 1.25 cm Sample 1.5.3: Convert 5.0 L to cm3 ( 1mL = 1cm3) 1000 mL 1 cm3 A = 5.0 L x x = 5.0 x 103 cm3 1L 1 mL  SAMPLE PROBLEMS: The height of a particular lighthouse is about 249 ft. Express this height in miles. 1 inch (in) = 2.54 cm 1 foot (ft) = 12 in 1 mile (mi) = 1.61 km A volume of a liquid substance is taken as 20.0 cc. Convert this volume in liters.  The deepest part of the Pacific Ocean is 5968 fathoms deep, what is this depth in meters? One fathom is exactly 6 feet 1 inch (in) = 2.54 cm 1 foot (ft) = 12 in 1 mile (mi) = 1.61 km ANS: 1.091 X 10^4 meters  A force of 2000 N acts on an area those measures 1 m x 2 m. What is the force per unit area in kN/m2? ANS: 1 kN/m2 One furlong is defined as one-eight of a mile. How many km. are there in a six-furlong race? The following relations are exact: 1 mile = 5280 feet, 12 in. = 1 ft., 1 in. = 2.54 cm. Express your answer in 2 decimal places. ANS: 1.21 km Add the following units of mass measurement and express answers in g. 0.456 kg, 965 mg, 1.83 dg, 0.537 g. ANS: 458 g  Physical Quantities Mass and Weight Mass is the quantity of matter that it contains. It is a constant quantity. The weight of a body is the force that gravity exerts on the body. Triple Beam Balance. A moderately accurate measurement is done. It is sensitive to 0.01 gram. ERROR IN READING Error due to incorrect PARALLAX ERROR positioning of eyes For accurate measurement the eye must always be placed vertically above the mark being read. For accuracy: To improve accuracy further , take several readings and use the average of these readings for better result.  Top Loading Balance. It is used for more accurate work. It is sensitive to 0.01gram to 0.0001 gram.  Spring Balance is used to measure the body's weight directly and is given in the units of force, the Newton (N). The spring is stretched until the force of gravity acting on the body is equal to the spring's backward pull.  Length Length is the distance covered by a line segment connecting two points. The standard unit of length in metric and SI units is the meter. Ruler. A ruler is a device used to measure more extended objects made up of plastic, wood, metal, etc. In centimeter measurement, there are ten divisions, and each is read as 1/10 or 0.1 cm. In inches measurement, there are 16 divisions, and each is read as 1/16 or 0.0625. Vernier Caliper A vernier caliper is an instrument that measures the internal diameter as well as the external diameter and distances of the object. This device takes more precise measurements than the regular ruler. The ruler has a sensitivity of 0.01 cm, while the vernier caliper has a sensitivity of 0.005 cm. https://www.youtube.com/watch?v=vkPlzmalvN4 ZERO ERROR IN VERNIER CALIPER PRACTICE: ANS: 0.28 cm PRACTICE: ANS: 0.57 cm PRACTICE: ANS: 0.76 cm PRACTICE: ANS: 0.60 cm ANS: 1.99 cm Zero error = +0.03 WHAT IS THE ACTUAL READING? ANS: 1.01 cm Zero error = -(0.10-.07) = -.03 WHAT IS THE ACTUAL READING? ANS: 1.40 cm WHAT IS THE ACTUAL READING? ANS: 0.27 cm Micrometer Caliper/Screw Gauge A micrometer screw gauge is a small measuring device that works on the screw principle and is used for measuring dimensions smaller than those measured by the vernier caliper. This device has a sensitivity of 0.001 mm and is more precise than using a vernier caliper. https://www.youtube.com/watch?v=StBc56ZifMs ZERO ERROR IN MICROMETER CALIPER PRACTICE: ANS: 12.18 mm ANS: 9.49 mm  ERROR ANS: -0.01 mm  ERROR ANS: +0.03 mm WHAT IS THE ACTUAL READING? ANS: 9.48 mm ANS: 12.45 mm ANS: 7.70 mm Area The area of plane figures or solids is the number of unit squares that can be contained within it. Volume Volume is defined in terms of standard length. Cubic meter (m3) is the standard SI-derived unit. It is the volume of a cube that is exactly one meter on edge. Another common unit of volume is liter. A liter is a non-SI unit of volume that is equal to 1000 mL or 1000 cc. Regular Solids ARCHIMEDES AND THE GOLD CROWN Irregular Solids The volume of an irregularly shaped object is determined by water displacement. Water displacement is done by placing the material in a graduated cylinder with a known volume of water. The material will displace a volume of water equal to the volume of the material. An increase in volume is the volume of the material.  Liquid Volume Measuring Devices Beakers and Flasks Graduated Cylinders Burette Pipette  Measuring Volume of Liquids When the liquid is poured into the cylinder, the volume is read from the scale on the side. To read the volume of the liquid accurately the base of the measuring cylinder must be placed on flat surface and eye must be leveled with the bottom of the meniscus. Meniscus: The surface of the liquid curves upwards at the point where it touches the inside of the cylinder. This curvature is called the meniscus. Density Density () is defined as the ratio of the samples’ mass to its volume. mass m density = or = volume V Density Density () is defined as the ratio of the samples’ mass to its volume. Density Density () is defined as the ratio of the samples’ mass to its volume. Sample 1.6.6: What is the mass of 10.0 mL cottonseed oil? The density of cottonseed oil is 0.926 g/mL at 20.0oC. Solution: Derive mass from the density equation and substitute the data m = m = xV V 0.926 g m= x 10.0 mL = 9.26 g mL Sample Problem 1.6.7: The water level in a graduated cylinder stands at 20.0 mL before and at 22.0 mL after a 5.4 g metal is submerged in water. Calculate the volume and density of the metal. Solution: The metal will displace a volume of water equal to the volume of the metal. Thus the increase in volume is the volume of the metal. Vmetal = Vwater+metal – Vwater = 22.0 mL – 20.0 mL = 2.0 mL m 5.4 g metal = = = 2.7 g/mL V 2.0 mL Specific Gravity (SG) denotes the ratio of a substance's density to a reference substance's density. Water is usually used as the reference substance for solids and liquids. Common reference substances used in specifying the specific gravity of gases are air and hydrogen.  density of solid or liquid Density of gas SG = density of water (1.00 g/mL) SG = Density of air (1.293 g/L) Specific gravity tells us how many times as heavy as a liquid, a solid, or a gas is compared to the reference material.  Sample 1.6.8: The specific gravity of mercury is 13.6 relative to water. What volume of Hg would have the same mass as 100.0 mL of water? Given: SGHg = 13.60 Vwater = 100.0 mL Required: VHg = ? Solution: m water = m Hg mHg  Hg VHg = V water SGHg =  water = VHg m water V water  V water 100.0 mL VHg= = = 7.35 mL SGHg 13.6  Temperature Temperature is the average kinetic energy of the particles in an object. Temperature is measured using a thermometer. The thermometer contains liquids, which expand when heated. Alcohol and mercury are commonly used in these thermometers. To convert oC to oF: TF = 9/5 (TC) + 32 where: TC = temperature in the Celsius scale TF= temperature in the Fahrenheit scale To convert oF to oC: TC = 5/9 (TF – 32) To convert oC to K: TK = TC + 273 To convert K to oC: TC = TK – 273 Practice Exercise 1.6: Answer the following in the right significant figures a.A sheet of aluminum foil has a total area of 1.000 ft2 and a mass of 3.636 g. What is the thickness of the foil in mm? (Al = 2.699 g/cc). Practice Exercise 1.6: Answer the following in the right significant figures b. Calculate the mass of each of the following: (a)a sphere of gold with radius of 10.0 cm,  = 19.3 g/cc; (b) a cube of platinum of edge length 0.0400 mm,  = 21.4 g/cc; (c) 50.0 cc of ethyl alcohol,  = 0.789 g/cc Practice Exercise 1.6: Answer the following in the right significant figures c. Normally the human body can endure a temperature of 105oF for a short time period without permanent damage to the brain and other vital organs. What is this temperature in degrees Celsius? In Kelvin?  PRECISION VS. ACCURACY GAME: GUESS MY AGE? 1.7 Precision and Accuracy Precision refers to how closely individual measurements agree with each other. It is usually expressed in terms of percent deviation We can calculate the % deviation from the formula Ave AD %D= x 100% M where: Ave AD = average absolute deviation M = mean or average of several measurements Absolute deviation (AD) is the difference between the measured value (Mo) and the mean (M) for the set of several measurements. In equation, AD = ǀMo – Mǀ Accuracy refers to how closely a measured value agrees with the correct value for a specific physical quantity. It is expressed in terms of percent error. We can calculate the % error from the formula Ave AE %E= x 100% MA where: Ave AE = average absolute error MA = accepted value Absolute error is the actual difference between the measured value and the accepted value. In equation, AE = ǀMo – MAǀ  PRECISION VERSUS ACCURACY not precise precise but precise and not accurate and not accurate accurate Percent Error can also be obtained by the absolute difference between the accepted value (or true value) and the average measurement (or average experiment value) divide by the accepted value. %E= ǀ TV – EV ǀ x 100% TV Sample Problem 1.7.1: Three students were asked to weigh 10.0 mL of distilled water. The results of two successive weighings by each student are: Student A Student B Student C 10.02 9.75 9.80 10.00 9.85 9.86 Average 10.01 9.80 9.83 value The true mass of distilled water is 10.00 g. Compare their measurements. Solution:  Ave AD 0.01 % DA = x 100 = x 100 = 0.1% M 10.01 Ave AD 0.05 % DB = x 100 = x 100 = 0.5% M 9.80 Ave AD 0.03 % DC = x 100 = x 100 = 0.3% M 9.83 Ave AE 0.01 % EA = x 100 = x 100 = 0.1% MA 10.00 Ave AE 0.20 % EB = x 100 = x 100 = 2.0% MA 10.00 Ave AE 0.17 % EC = x 100 = x 100 = 1.7% MA 10.00 ǀ TV – EV ǀ ǀ 10.00 – 10.01 ǀ x 100 = 0.1% % EA = x 100 = TV 10.00 ǀ TV – EV ǀ ǀ 10.00 – 9.80 ǀ % EB = x 100 = x 100 = 2.0% TV 10.00 ǀ TV – EV ǀ ǀ 10.00 – 9.83 ǀ % EC = x 100 = x 100 = 1.7% TV 10.00  Student C's results are more precise than Student B's results, but neither set results are very accurate. Student A’s results are not only the most precise but also the most accurate. Practice Exercise 1.7: The mass of an unknown substance is 2.86 g. Which of the following sets of measurement represents the value with both accuracy and precision? (a)1.78 g, 1.80 g, 1.76g, 1.81 g; (b) 2.85 g, 2.86 g, 2.84, 2.81 g: (c) 1.98 g, 2.02 g, 1.96 g, 2.01g (d) 2.81 g, 1.98 g, 2.40 g, 2.78 g