4.4: Ideal Gas Model

Construct Definitions and Relationships

1. The Ideal Gas Model is the entire set of ideas related to gases that scientists regularly use as whenever they reason about phenomena involving gases. Whether gases involved in the particular phenomenon “obey” the ideal gas law or not, the starting point for thinking about gases is the Ideal Gas Model. It embodies the ideas that gases are composed of extremely small particles with a lot of space between them that interact only like perfect billiard balls when they happen to collide with each other.

2. The Ideal Gas Model is so widely use by practically all scientists because on the one hand the mathematical relationships between the variables are straightforward and easy to reason with, and on the other hand, gas properties appearing in the model are empirically meaningful and can be rigorously determined.

3. As represented in terms of the “ideal gas law” it is a model for connecting the microscopic way of thinking about gases to the macroscopic way of thinking about gases. This is strikingly seen in the two equivalent ways the model is written:

\( PV = nRT \) and \( PV = Nk_{B}T \),

where n is the number of moles and R is the gas constant (both macroscopic constructs) and N is the number of molecules and k_B is the Boltzmann’s constant (both microscopic constructs).

4. The temperature term forms a bridge between the particle model of thermal energy and macroscopic thermodynamics. The temperature connects directly to the meaning of thermal energy through the fundamental relationship of temperature as a measure of the random thermal motion in any energy mode in thermal equilibrium at temperature T: E_thermal/mode = (1/2)k_BT, while at the same time, temperature is seen to be directly proportional to the pressure of a gas.

5. When combined with the particle model of thermal energy, E_thermal (total) = (total number of active modes) ´ (1/2)k_BT, the Ideal Gas Model provides substantial insight into what the fundamental properties of an ideal gas are from a PV perspective and how they differ from an internal energy perspective, as well as providing a bridge between the two.

6. The basic concept of heat capacity at constant pressure (Cp = Cv + PdV/dT),
when applied to an ideal gas gives the fundamental result that the molar heat capacity at constant pressure of any ideal gas is greater than the heat capacity at constant volume simply by the additive factor R.

Algebraic Relationships

Ideal Gas Law:

PV = nRT or PV = N k_BT

R = 8.314 J / K·mol = N_Ak_B

n is number of moles

k_B = 1.381 x 10^-23 J/K

N is number of molecules

Relationships that are connected by the thermodynamic temperature, T

\( PV = nRT \text{ or } PV = N k_{B}T \)

\( dU = TdS – PdV \)

\( C_{v} = dE_{th}/dT \)

\( E_{thermal} / \text{mode} = \frac{1}{2} k_{B} T \)

\( E_{thermal} \text{(total)} = \text{(total number of active modes)} \times´ \frac{1}{2} k_{B} T \)

Heat Capacity for an Ideal Gas at constant pressure

Our general result for the heat capacity at constant pressure is

\(C_{p} = C_{v} + P dV/dT \)

We can directly calculate dV/dT from the equation of state for an ideal gas. Rearranging and differentiating PV = nRT (or for one mole, PV = RT), we get

dV/dT = R/P (remember P is constant)

Then, the expression for the molar heat capacity at constant pressure for an ideal gas becomes

\( c_{pm} = c_{vm} + R \)

For an ideal gas, the molar heat capacity at constant pressure is larger than at constant volume by exactly the value R. This is true for any ideal gas, whether monatomic, diatomic, or polyatomic, because the ideal gas law doesn’t depend on intra-atomic motions. Looking back at the table of experimentally determined heat capacities at the beginning of this chapter, we see indeed that the molar heat capacity measured at constant pressure is larger than the constant volume heat capacity by one R. Our model of matter does indeed work pretty well for gases.