Lecture 10 Gases & Kinetic Theory

17-6 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. Figure 17-13: Pressure vs. volume of a fixed amount of gas at a constant temperature, showing the inverse relationship as given by Boyle’s law: as the pressure decreases, the volume increases.

17-6 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. Extrapolating, the volume becomes zero at −273.15°C; this temperature is called absolute zero. Figure 17-14. Volume of a fixed amount of gas as a function of (a) Celsius temperature, and (b) Kelvin temperature, when the pressure is kept constant.

17-6 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.

17-7 The Ideal Gas Law We can combine the three relations just derived into a single relation: What about the amount of gas present? If the temperature and pressure are constant, the volume is proportional to the amount of gas: Figure 17-15. Blowing up a balloon means putting more air (more air molecules) into the balloon, which increases its volume. The pressure is nearly constant (atmospheric) except for the small effect of the balloon’s elasticity.

17-7 The Ideal Gas Law A mole (mol) is defined as the number of grams of a substance that is numerically equal to the molecular mass of the substance: 1 mol H2 has a mass of 2 g. 1 mol Ne has a mass of 20 g. 1 mol CO2 has a mass of 44 g. The number of moles in a certain mass of material:

17-7 The Ideal Gas Law We can now write the ideal gas law: where n is the number of moles and R is the universal gas constant.

17-8 Problem Solving with the Ideal Gas Law Standard temperature and pressure (STP): T = 273 K (0°C) P = 1.00 atm = 1.013 × 105 N/m2 = 101.3 kPa. Example 17-10: Volume of one mole at STP. Determine the volume of 1.00 mol of any gas, assuming it behaves like an ideal gas, at STP. Solution: Substituting gives V = 22.4 x 10-3 m3, or 22.4 liters.

17-8 Problem Solving with the Ideal Gas Law Volume of 1 mol of an ideal gas is 22.4 L If the amount of gas does not change: Always measure T in kelvins P must be the absolute pressure

17-8 Problem Solving with the Ideal Gas Law Example 17-13: Check tires cold. An automobile tire is filled to a gauge pressure of 200 kPa at 10°C. After a drive of 100 km, the temperature within the tire rises to 40°C. What is the pressure within the tire now? Solution: The volume stays the same, so the ratio of the pressure to the temperature does too. This gives a gauge pressure of 232 kPa.

Chapter 18 Kinetic Theory of Gases Chapter opener. In this winter scene in Yellowstone Park, we recognize the three states of matter for water: as a liquid, as a solid (snow and ice), and as a gas (steam). In this Chapter we examine the microscopic theory of matter as atoms or molecules that are always in motion, which we call kinetic theory. We will see that the temperature of a gas is directly related to the average kinetic energy of its molecules. We will consider ideal gases, but we will also look at real gases and how they change phase, including evaporation, vapor pressure, and humidity.

Units of Chapter 18 The Ideal Gas Law and the Molecular Interpretation of Temperature Distribution of Molecular Speeds

18-1 The Ideal Gas Law 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

18-1 The Ideal Gas Law 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 Figure 1801. (a) Molecules of a gas moving about in a rectangular container. (b) Arrows indicate the momentum of one molecule as it rebounds from the end wall.

18-1 The Ideal Gas Law and the Molecular Interpretation of Temperature The averages of the squares of the speeds in all three directions are equal: So the pressure is:

18-1 The Ideal Gas Law 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.

18-1 The Ideal Gas Law and the Molecular Interpretation of Temperature Example 18-1: Molecular kinetic energy. What is the average translational kinetic energy of molecules in an ideal gas at 37°C? Solution: Substitution gives K = 6.42 x 10-21 J.

18-1 The Ideal Gas Law and the Molecular Interpretation of Temperature We can now calculate the average speed of molecules in a gas as a function of temperature:

18-1 The Ideal Gas Law and the Molecular Interpretation of Temperature Example 18-2: Speeds of air molecules. What is the rms speed of air molecules (O2 and N2) at room temperature (20°C)? Solution: The speeds are found from equation 18-5, and are different for oxygen and nitrogen (it’s the kinetic energies that are the same). Oxygen: 480 m/s. Nitrogen: 510 m/s.

18-1 The Ideal Gas Law and the Molecular Interpretation of Temperature Conceptual Example 18-3: Less gas in the tank. A tank of helium is used to fill balloons. As each balloon is filled, the number of helium atoms remaining in the tank decreases. How does this affect the rms speed of molecules remaining in the tank? Solution: If the temperature remains the same, the rms speed does not change.

18-1 The Ideal Gas Law and the Molecular Interpretation of Temperature Example 18-4: Average speed and rms speed. Eight particles have the following speeds, given in m/s: 1.0, 6.0, 4.0, 2.0, 6.0, 3.0, 2.0, 5.0. Calculate (a) the average speed and (b) the rms speed. Solution: The average is 3.6 m/s and the rms is 4.0 m/s.

18-2 Distribution of Molecular Speeds The molecules in a gas will not all have the same speed; their distribution of speeds is called the Maxwell distribution: Figure 18-2. Distribution of speeds of molecules in an ideal gas. Note that vav and vrms are not at the peak of the curve. This is because the curve is skewed to the right: it is not symmetrical. The speed at the peak of the curve is the “most probable speed,” vp .

18-2 Distribution of Molecular Speeds The Maxwell distribution depends only on the absolute temperature. This figure shows distributions for two different temperatures; at the higher temperature, the whole curve is shifted to the right. Figure 18-3: Distribution of molecular speeds for two different temperatures.

19-2 Internal Energy The sum total of all the energy of all the molecules in a substance is its internal (or thermal) energy. Temperature: measures molecules’ average kinetic energy Internal energy: total energy of all molecules Heat: transfer of energy due to difference in temperature

19-2 Internal Energy Eint = 3/2 nRT Internal energy of an ideal (atomic) gas: But since we know the average kinetic energy in terms of the temperature, we can write: Eint = 3/2 nRT

19-2 Internal Energy If the gas is molecular rather than atomic, rotational and vibrational kinetic energy need to be taken into account as well. Figure 19-2. Besides translational kinetic energy, molecules can have (a) rotational kinetic energy, and (b) vibrational energy (both kinetic and potential).

19-8 Degrees of Freedom (dof) The number of independent ways molecules can possess energy. Monoatomic gas (eg Argon) – 3 dof (due to translational velocity in the x, y & z direction) Diatomic gas (eg O2)– 5 dof (3 translational, 2 rotational); 7 dof at high temperature. (2 additional dof from vibrational motion) More complex gas molecules – more degrees of freedom At low temperature nearly all molecules have only translational kinetic energy

19-8 Molar Specific Heats for Gases, and the Equipartition of Energy For a gas consisting of more complex molecules (diatomic or more), the molar specific heats increase. This is due to the extra forms of internal energy that are possible (rotational, vibrational). Figure 19-15. A diatomic molecule can rotate about two different axes. Figure 19-16. A diatomic molecule can vibrate, as if the two atoms were connected by a spring. Of course they are not connected by a spring; rather they exert forces on each other that are electrical in nature, but of a form that resembles a spring force.

19-8 Equipartition of Energy The equipartition theorem states that the total internal energy is shared equally among the active degrees of freedom, each accounting for ½ kT. Figure 19-17. Molar specific heat CV as a function of temperature for hydrogen molecules (H2). As the temperature is increased, some of the translational kinetic energy can be transferred in collisions into rotational kinetic energy and, at still higher temperature, into vibrational kinetic energy. [Note: H2 dissociates into two atoms at about 3200 K, so the last part of the curve is shown dashed.]

Summary of Chapter 18 The average kinetic energy of molecules in a gas is proportional to the temperature. Evaporation occurs when the fastest moving molecules escape from the surface of a liquid.

Summary of Chapter 19 Internal energy, Eint, refers to the total energy of all molecules in an object. For an ideal monatomic gas,