Week 09 - Gas Properties - 2023/2024 PDF

Gas Properties SIO1001 Sem2 2023/2024  Chapter 13 Temperature and Kinetic Theory Contents Gentle introduction- Atomic Theory of Avogadro’s Law Matter Ideal Gas Equation Temperature, Thermometers & Thermometry...

Gas Properties SIO1001 Sem2 2023/2024  Chapter 13 Temperature and Kinetic Theory Contents Gentle introduction- Atomic Theory of Avogadro’s Law Matter Ideal Gas Equation Temperature, Thermometers & Thermometry Density of Gas Thermal Equilibrium and the Zeroth Law Molecular Mass of Thermodynamics Partial Pressure The Gas Laws and Absolute Temperature Molecular Kinetic Theory and the Boyle’s Law Molecular Interpretation of Temperature Charles' Law Distribution of Molecular Speeds Gay-Lussac’s Law Real Gases and Changes of Phase The Ideal Gas Law Diffusion & Effusion Combined Gas Law Gentle Introduction  Properties of Gases 1. Gas molecules have mass 2. The molecules move randomly 3. Gases take shape & volume of their container 4. Gases are compressible (because they are separated by great distances) 5. Gases move through regions easily  “Diffusion”  perfume, skunks! 6. Gases exert pressure 7. Etc...  Atomic Theory of Matter Atomic and molecular masses are measured in unified atomic mass units (u). This unit is defined so that the carbon-12 atom has a mass of exactly 12.0000 u. Expressed in kilograms: 1 u = 1.6605 × 10−27 kg Brownian motion is the jittery motion of tiny flecks in water; these are the result of collisions with individual water molecules.  Temperature and Thermometers Temperature is a measure of how hot or cold something is. Most materials expand when heated.  Temperature and Thermometers Thermometers are instruments designed to measure temperature. In order to do this, they take advantage of some property of matter that changes with temperature. Early thermometers:  Temperature and Thermometers Common thermometers used today include the liquid-in-glass type and the bimetallic strip. Temperature and Thermometers Temperature is generally measured using either the Fahrenheit or the Celsius scale. The freezing point of water is 0°C, or 32°F; the boiling point of water is 100°C, or 212°F.  Thermal Equilibrium (Zeroth Law of Thermodynamics) Two objects placed in thermal contact will eventually come to the same temperature. When they do, we say they are in thermal equilibrium. The zeroth law of thermodynamics says that if two objects are each in equilibrium with a third object, they are also in thermal equilibrium with each other. Gas Laws  The Gas Laws and Absolute Temperature The relationship between the volume, pressure, temperature, and mass of a gas is called an equation of state. We will deal here with gases that are not too dense. Boyle’s Law: the volume of a given amount of gas is inversely proportional to the pressure as long as the temperature is constant. V ∝ 1/P  The Gas Laws and Absolute Temperature The volume is linearly proportional to the temperature, as long as the temperature is somewhat above the condensation point and the pressure is constant: V ∝ T. Extrapolating, the volume becomes zero at −273.15°C; this temperature is called absolute zero.  The Gas Laws and Absolute Temperature The concept of absolute zero allows us to define a third temperature scale—the absolute, or Kelvin, scale. This scale starts with 0 K at absolute zero, but otherwise is the same as the Celsius scale. Therefore, the freezing point of water is 273.15 K, and the boiling point is 373.15 K. Finally, when the volume is constant, the pressure is directly proportional to the temperature: P ∝T. The Gas Laws Learning Goals: We will review Boyle’s, Charles’ and Gay- Lussac’s Laws relating T, P and/or V and be able to calculate unknown values using the equations derived from these laws, as well as the combined gas law. Boyle’s Law  Intro to Boyle’s Law Imagine that you hold the tip of a syringe on the tip of your finger so no gas can escape. Now push down on the plunger of the syringe. What happens to the volume in the syringe? What happens to the pressure the gas is exerting in the syringe? Boyle’s Law  Boyle’s Law The pressure and volume of a gas are inversely proportional (as one increases, the other decreases, and vice versa – at constant mass & temp P V  Boyle’s Law Boyle’s Law leads to the mathematical expression: *Assuming temp is constant P1V1=P2V2 Where P1 represents the initial pressure V1 represents the initial volume, And P2 represents the final pressure V2 represents the final volume  Q1 A weather balloon with a volume of 2000L at a pressure of 96.3 kPa rises to an altitude of 1000m, where the atmospheric pressure is measured to be 60.8kPa. Assuming there is no change in the temperature or the amount of gas, calculate the weather balloon’s final volume.  Q2 Atmospheric pressure on the peak of Kilimanjaro can be as low as 0.20 atm. If the volume of an oxygen tank is 10.0L, at what pressure must the tank be filled so the gas inside would occupy a volume of 1.2 x 103L at this pressure? Charles’ Law  Intro to Charles’ Law Imagine that you put a balloon filled with gas in liquid nitrogen What is happening to the temperature of the gas in the balloon? What will happen to the volume of the balloon?  Charles’ Law The volume and absolute temperature (K) of a gas are directly proportional (an increase in temp leads to an increase in volume) – at constant mass & pressure Charles’ Law leads to the mathematical expression: – *Assuming pressure remains constant V T Charles’ Law  Q3 A birthday balloon is filled to a volume of 1.5L of helium gas in an air-conditioned room at 293K. The balloon is taken outdoors on a warm day where the volume expands to 1.55L. Assuming the pressure and the amount of gas remain constant, what is the air temperature outside in Celsius?  Q4 A beach ball is inflated to a volume of 25L of air at 15oC. During the afternoon, the volume increases by 1L. What is the new temperature outside? Gay-Lussac’s Law  Intro to Gay-Lussac’s Law Imagine you have a balloon inside a container that ensures it has a fixed volume. You heat the balloon. What is happening to the temp of the gas inside the balloon? What will happen to the pressure the gas is exerting on the balloon?  Gay-Lussac’s Law The pressure and absolute temperature (K) of a gas are directly proportional (as temperature rises, so does pressure) – at constant mass & volume Gay-Lussac’s Law leads to the mathematical expression as below: *Assuming volume remains constant P 𝑃1 𝑃2 = T 𝑇1 𝑇2  Q5 The pressure of the oxygen gas inside a canister with a fixed volume is 5.0atm at 15oC. What is the pressure of the oxygen gas inside the canister if the temperature changes to 263K? Assume the amount of gas remains constant.  Q6 The pressure of a gas in a sealed canister is 350.0kPa at a room temperature of 15oC. The canister is placed in a refrigerator that drops the temperature of the gas by 20K. What is the new pressure in the canister? Combined Gas Law  Combined Gas Law By combining Boyle’s, Charles’ and Gay Lussac’s Laws, the following equation is derived: P1V1 P2V2 = T1 T2  Q7 A gas occupies 7.84 cm3 at 71.8 kPa & 25°C. Find its volume at STP.  Q8: Any Combination Questions  a) A gas occupies 473 cm 3 at 36°C. Find its volume at 94°C b) A gas’ pressure is 765 torr at 23°C. At what temperature will the pressure be 560. torr Avogadro’s Law  Avogadro’s Law For an ideal gas, the volume (V) and moles of gas (n) are directly proportional if the temperature and pressure are constant. Ideal Gas Ideal-Gas Equation So far, we’ve seen that V  1/P (Boyle’s law) V  T (Charles’s law) V  n (Avogadro’s law) P  T (Gay-Lussac’s law) Combining these, we get nT V P Ideal-Gas Equation The relationship nT V P then becomes nT V=R P or PV = nRT © 2009, Prentice-Hall, Inc. Ideal-Gas Equation The constant of proportionality is known as R, the gas constant. Densities of Gases Densities of Gases If we divide both sides of the ideal-gas equation by V and by RT, we get n P = V RT Densities of Gases We know that moles  molecular mass = mass n=m So, multiplying both sides by the molecular mass ( ) gives m P = V RT Densities of Gases Mass  volume = density So, m P d= = V RT Note: One only needs to know the molecular mass, the pressure, and the temperature to calculate the density of a gas. Molecular Mass We can manipulate the density equation to enable us to find the molecular mass of a gas: P d= RT Becomes dRT = P Partial Pressure – Delton’s Law Partial Pressure – Delton’s Law For a mixture of gases in a container, PTotal = P1 + P2 + P3 +... The total pressure exerted is the sum of the pressures that each gas would exert if it were alone. Partial Pressure – Delton’s Law P1 P2 PTOTAL= P1 + P2  Q9 Consider the following apparatus containing helium in both sides at 45°C. Initially the valve is closed. After the valve is opened, what is the pressure of the helium gas? 2.50 atm 4.50 atm 8.00 L 2.50 L  Q10 Consider the following apparatus containing helium in both sides at 45°C. Initially the valve is closed. After the valve is opened, what is the pressure of the helium gas? 2.00 atm 3.00 atm 9.00 L 3.00 L  Q11 27.4 L of oxygen gas at 25.0°C and 1.30 atm, and 8.50 L of helium gas at 25.0°C and 2.00 atm were pumped into a tank with a volume of 5.81 L at 25°C. Calculate the new partial pressure of oxygen. 6.13 atm Calculate the new partial pressure of helium. 2.93 atm Calculate the new total pressure of both gases. 9.06 atm Molecular Kinetic Theory Kinetic Theory of Molecule All matter is made of atoms or molecules that are in constant motion These particles contain energy The movement of these particles is random  Kinetic Theory and the Molecular Interpretation of Temperature Assumptions of kinetic theory: large number of molecules, moving in random directions with a variety of speeds molecules are far apart, on average molecules obey laws of classical mechanics and interact only when colliding collisions are perfectly elastic Kinetic Theory and the Molecular Interpretation of Temperature The force exerted on the wall by the collision of one molecule is Then the force due to all molecules colliding with that wall is Kinetic Theory and the Molecular Interpretation of Temperature The averages of the squares of the speeds in all three directions are equal: So the pressure is: (13-6) Kinetic Theory and the Molecular Interpretation of Temperature Rewriting, so The average translational kinetic energy of the molecules in an ideal gas is directly proportional to the temperature of the gas. Kinetic Theory and the Molecular Interpretation of Temperature We can invert this to find the average speed of molecules in a gas as a function of temperature: Distribution of Molecular Speeds These two graphs show the distribution of speeds of molecules in a gas, as derived by Maxwell. The most probable speed, vP, is not quite the same as the rms speed. As expected, the curves shift to the right with temperature. Real Gas Real Gases and Changes of Phase The curves here represent the behavior of the gas at different temperatures. The cooler it gets, the farther the gas is from ideal. In curve D, the gas becomes liquid; it begins condensing at (b) and is entirely liquid at (a). The point (c) is called the critical point.  Real Gases and Changes of Phase Below the critical temperature, the gas can liquefy if the pressure is sufficient; above it, no amount of pressure will suffice.  Real Gases and Changes of Phase A PT diagram is called a phase diagram; it shows all three phases of matter. The solid-liquid transition is melting or freezing; the liquid-vapor one is boiling or condensing; and the solid-vapor one is sublimation. Phase diagram of water  Real Gases and Changes of Phase The triple point is the only point where all three phases can coexist in equilibrium. Phase diagram of carbon dioxide Diffusion & Effusion  Diffusion Even without stirring, a few drops of dye in water will gradually spread throughout. This process is called diffusion.  Diffusion Diffusion occurs from a region of high concentration towards a region of lower concentration.  Diffusion The rate of diffusion is given by: (13-10) In this equation, D is the diffusion constant.  Effusion Diffusion – the mixing of gases. Effusion – describes the passage of a gas through a tiny orifice into an evacuated chamber. Rate of effusion measures the speed at which the gas is transferred into the chamber. Copyright © Cengage Learning. All rights reserved 74 Effusion 75 Summary Gentle introduction- Atomic Theory of Avogadro’s Law Matter Ideal Gas Equation Temperature, Thermometers & Thermometry Density of Gas Thermal Equilibrium and the Zeroth Law Molecular Mass of Thermodynamics Partial Pressure The Gas Laws and Absolute Temperature Molecular Kinetic Theory and the Boyle’s Law Molecular Interpretation of Temperature Charles' Law Distribution of Molecular Speeds Gay-Lussac’s Law Real Gases and Changes of Phase The Ideal Gas Law Diffusion & Effusion Combined Gas Law  Practical Questions Chapter 15 Applied Physics by Gunderson 10th Ed Chapter 15 Applied Physics by Gunderson 10th Ed END