The Kinetic Theory of Gases Chapter 19 The Kinetic Theory of Gases From the macro-world to the micro-world Key contents: Ideal gases Pressure, temperature, and the RMS speed Molar specific heats Adiabatic expansion of ideal gases

One mole is the number of atoms in a 12 g sample of carbon-12. 19.2 Avogadro’s Number Italian scientist Amedeo Avogadro (1776-1856) suggested that all gases occupy the same volume under the condition of the same temperature, the same pressure, and the same number of atoms or molecules. => So, what matters is the ‘number’ . One mole is the number of atoms in a 12 g sample of carbon-12. The number of atoms or molecules in a mole is called Avogadro’s Number, NA. If n is the number of moles contained in a sample of any substance, N is the number of molecules, Msam is the mass of the sample, m is the molecular mass, and M is the molar mass, then

The equation of state of a dilute gas is found to be 19.3: Ideal Gases The equation of state of a dilute gas is found to be Here p is the pressure, n is the number of moles of gas present, and T is its temperature in kelvins. R is the gas constant that has the same value for all gases. Or equivalently, Here, k is the Boltzmann constant, and N the number of molecules. (# The ideal gas law can be derived from the Maxwell distribution; see slides below.)

19.3: Ideal Gases; Work Done by an Ideal Gas

Example, Ideal Gas Processes

Example, Work done by an Ideal Gas

19.4: Pressure, Temperature, and RMS Speed The momentum delivered to the wall is +2mvx Considering , we have Defining , we have Comparing that with , we have The temperature has a direct connection to the RMS speed squared.

Translational Kinetic Energy

19.4: RMS Speed

Example:

19.7: The Distribution of Molecular Speeds Maxwell’s law of speed distribution is: The quantity P(v) is a probability distribution function: For any speed v, the product P(v) dv is the fraction of molecules with speeds in the interval dv centered on speed v. Fig. 19-8 (a) The Maxwell speed distribution for oxygen molecules at T =300 K. The three characteristic speeds are marked.

Example, Speed Distribution in a Gas:

Example, Different Speeds

19.8: Molar Specific Heat of Ideal Gases: Internal Energy The internal energy Eint of an ideal gas is a function of the gas temperature only; it does not depend on any other variable. For a monatomic ideal gas, only translational kinetic energy is involved.

19.8: Molar Specific Heat at Constant Volume where CV is a constant called the molar specific heat at constant volume. But, Therefore, With the volume held constant, the gas cannot expand and thus cannot do any work. # When a confined ideal gas undergoes temperature change DT, the resulting change in its internal energy is A change in the internal energy Eint of a confined ideal gas depends on only the change in the temperature, not on what type of process produces the change.

19.8: Molar Specific Heat at Constant Pressure where Cp is a constant called the molar specific heat at constant pressure. This Cp is greater than the molar specific heat at constant volume CV, since for the same internal energy change, more heat is needed to provide work.

Example, Monatomic Gas:

Molar specific heats at 1 atm, 300K CV (J/mol/K) CP-CV (J/mol/K) g=CP/CV monatomic 1.5R=12.5 R=8.3 He 12.5 8.3 1.67 Ar diatomic 2.5R=20.8 H2 20.4 8.4 1.41 N2 20.8 1.40 O2 21.0 Cl2 25.2 8.8 1.35 polyatomic 3.0R=24.9 CO2 28.5 8.5 1.30 H2O(100°C) 27.0 1.31

19.9: Degrees of Freedom and Molar Specific Heats Every kind of molecule has a certain number f of degrees of freedom, which are independent ways in which the molecule can store energy. Each such degree of freedom has associated with it —on average — an energy of ½ kT per molecule (or ½ RT per mole). This is equipartition of energy. Recall that

CV (J/mol/K) CP-CV (J/mol/K) g=CP/CV monatomic 1.5R=12.5 R=8.3 He 12.5 8.3 1.67 Ar diatomic 2.5R=20.8 H2 20.4 8.4 1.41 N2 20.8 1.40 O2 21.0 Cl2 25.2 8.8 1.35 polyatomic 3.0R=24.9 CO2 28.5 8.5 1.30 H2O(100°C) 27.0 1.31

Example, Diatomic Gas:

# Oscillations are excited with 2 degrees of freedom 19.10: A Hint of Quantum Theory A crystalline solid has 6 degrees of freedom for oscillations in the lattice. These degrees of freedom are frozen (hidden) at low temperatures. # Oscillations are excited with 2 degrees of freedom (kinetic and potential energy) for each dimension. # Hidden degrees of freedom; minimum amount of energy # Quantum Mechanics is needed.

19.11: The Adiabatic Expansion of an Ideal Gas with Q=0 and dEint=nCVdT , we get: From the ideal gas law, and since CP-CV = R, we get: With g = CP/CV, and integrating, we get: Finally we obtain:

19.11: The Adiabatic Expansion of an Ideal Gas

19.11: The Adiabatic Expansion of an Ideal Gas, Free Expansion A free expansion of a gas is an adiabatic process with no work or change in internal energy. Thus, a free expansion differs from the adiabatic process described earlier, in which work is done and the internal energy changes. In a free expansion, a gas is in equilibrium only at its initial and final points; thus, we can plot only those points, but not the expansion itself, on a p-V diagram. Since ΔEint =0, the temperature of the final state must be that of the initial state. Thus, the initial and final points on a p-V diagram must be on the same isotherm, and we have Also, if the gas is ideal,

Example, Adiabatic Expansion:

Four Gas Processes for an Ideal Gas

Homework: Problems 13, 24, 38, 52, 59