Problem 2.5.1

If pressure, volume and temperature relationship for nitrogen is given by Redlich-Kwong equation, then determine the value of specific volume, compressibility factor and fugacity coefficient at 100 bars and 250 degrees Kelvin.

Critical pressure of nitrogen is 33.5 atmospheres, and critical temperature is 125.95 degrees Kelvin.

Solution:

The two-constant equation of Redlich-Kwong is given as: P is equal to RT over (V minus b) minus ‘a’ over [T^1/2 V (V plus b)]. The constants for use in an equation of state are best found by a least-squares fit of the equation to experimental PVT data. However, such data are frequently not available. It is noted that the PV isotherm passing through the critical point exhibits a horizontal inflection at this point. Thus the critical isotherm has zero slope and zero curvature at the critical point. Mathematically, (partial P over partial V) is equal to partial² P over partial V² at critical temperature is equal to zero. The results of this procedure are:

‘a’ is equal to 0.4278 R²T_C^2.5 over [64 P_c] and ‘b’ is equal to 0.0867 RT_c over P_c. It should be noted that equations of state that are cubic in V have either one or three real roots that satisfy the equation at each T and P. Where three real roots exist, the largest is a vapor volume and the smallest is a liquid volume. The middle root has not significance.

Since these values are calculated from data for a single point, the critical point, they will not likely be the best values. However, they are reasonable values which can almost always be determined, for critical temperatures and pressures are known for a very large number of substances.

The Redlich-Kwong equation is often written in an alternate form as Z is equal to one over [one minus h] minus A over B (h over [one plus h]), where h is equal to b over V or BP over Z. In these relations, B is equal to b over RT and [A over B] is equal to ‘a’ over [b RT^1.5].

Gas constant is 8.32 Pascal cubic meter per mole per degree Kelvin. The value of ‘a’ is obtained as 1.551 and ‘b’ is 2.675 times 10^-5.

P is equal to RT over (V minus b) minus ‘a’ over [T^1/2 V (V plus b)]. Plugging in the values of ‘a’, ‘b’, and pressure (10⁵ Pa), a trial and error approach is needed. Initial guess is the ideal volume as 2.079 times 10^-4 cubic meter. The final converged value is 1.97 times 10^-4 cubic meter.

Compressibility factor, Z, is PV over RT and is equal to 0.946.

Perry’s equation 4.321 gives a mathematical relationship to find the fugacity coefficient. The variable h is given by b over V and is equal to 0.136. Now, log of phi is calculated as minus 0.077. This gives fugacity coefficient of 0.926.