Ideal Gas Properties

What is an ideal gas?

An ideal gas is a special case of a pure substance in the vapor phase. In fact, it is an idealization of such a substance. The following graph of experimental data for nitrogen shows the variation of the Pv product (the "virial") with both temperature and pressure. Note that the volume in this graph is the molar volume, which is the total volume divided by the number of moles.

In general, the Pv data shown in this graph can be represented mathematically by a power series in P, in which the coefficients are functions of T, as follows:

When the Pv data are extrapolated to zero pressure (P = 0) and then are plotted against the absolute temperature in kelvins, T(K), the data points lie along a straight line, so that

where

is the slope of the line.

It has been found that the data points for all gases lie on the same straight line when the virial is expressed in molar terms. The equation is usually called the "ideal gas equation of state" and is called the "universal" gas constant. It can be seen that for an ideal gas, the equation for the virial includes only the first term in the power series. When the virial is expressed in specific terms, the corresponding equation is

Pv = RT

where v is the specific volume and R is the "specific" gas constant.

There are other equations of state, including those for internal energy and enthalpy. For any gas, u(P,T) and h(P,T) can also be expressed mathematically as power series in P in which the coefficients are functions of T. For an ideal gas, these equations of state also include only the first, temperature-dependent (only), term:

Thus, a real gas is said to "behave as" an ideal gas, when the pressure is low enough that the higher order pressure-dependent terms in the equations of state can be neglected.

Ideal gas equations of state

The thermodynamic properties of most interest are the internal energy, enthalpy, and entropy. "Ideal gas equations of state" relate these properties to the properties pressure, temperature, and specific volume.

The equations of state are usually written in terms of certain derivatives, called the specific heats. The "isochoric specific heat" or "specific heat at constant volume" is defined as

c_v = du/dT
and the "isobaric specific heat" or "specific heat at constant pressure" is defined as c_p = dh/dT.

The specific heats are functions of temperature. The following graph shows the variation of the isobaric specific heat, c_p, with temperature. The curves for the isochoric specific heat, c_v, have the same shape because

c_p - c_v = R.

In this graph, the specific heats are given in molar units. For all properties, the value of the specific property can be obtained from the value of the molar property by dividing by the molar mass (molecular weight) M of the gas.

Note that the specific heats are constant for monatomic gases and vary more strongly with temperature for triatomic gases than for diatomic gases. It can be seen that for low temperatures,the specific heat values are nearly constant. For some gases, they are constant at room temperature (300 K). For higher temperatures, it is often a useful approximation to assume the specific heats to be constant, taking the values to be the mean values for the temperature range of interest. One must be careful, of course, that the temperature range is not too large.

In terms of the specific heats, the ideal gas equations of state for internal energy and enthalpy are expressed in terms of integrals of the specific heats:

where u_o(T_r) and h_o(T_r) are "reference" values of the properties at the "reference" temperature T_r.

For the entropy, the fundamental equation,

T ds = du + P dv

or its equivalent,

T ds = dh - v dP

can be solved for ds and then integrated, with the following results:

Special equations for constant specific heats

For constant specific heats, the equations of state for internal energy, enthalpy, and entropy reduce to the following simple forms:

Ideal gas property graphs

Since an ideal gas is a special case of a pure substance, the state is specified by giving the values of two properties. All other properties can then be found from the equations of state. Just as for a general pure substance, on a three dimensional plot of any three properties, all states of equilibrium of an ideal gas will correspond to points on a surface. For example, the following graph shows an isometric view of the P-v-T surface of an ideal gas.

Of course, the P-v-T surface for an ideal gas is much simpler that that for the general case of a pure substance.

The next two graphs show two-dimensional representations of the P-v-T surface, using constant-property lines to indicate the shaper of the surface.

Compare this with the P-T diagram for a pure substance.

Compare with the P-v diagram of a pure substance.

Compare with the T-s diagram for a pure substance.