The ideal gas equation of state is very simple, but its range of applicability i

Question

The ideal gas equation of state is very simple, but its range of applicability is limited. A more accurate but complicated equation is the Van der Waals equation of state given by

where a and b are constants depending on critical pressure and temperatures of the gas.

Calculate the coefficient of compressibility of nitrogen gas at T = 175 K and V=0.00375m3/kg, assuming the nitrogen to obey the Van der Waals equation of state.

And compare the result with the ideal gas value. (a=0.175 m6kPa/kg2 and b=0.00138 m3/kg for the given conditions. The experimentally measured pressure of nitrogen is 10,000 kPa).

Explanation / Answer

Coeff of Compressibility, k = -1/v (dv/dP)T

Taking derivative of Van der Waals equation with respect to v at constant T we get

(dP/dv)T = RT ln (v-b) + (2a / v3)

Putting values, (dP/dv)T = 0.297*175 ln (0.00375 - 0.00138) + (2*0.175 / 0.003753)

(dP/dv)T = 6636723 kPa / m3

k = -(1/v) / 6636723

k = -(1/0.00375) / 6636723

k = -4.018*10-5 /kPa

For ideal gas, P = RT/v

(dP/dv)T = RT (ln v)

= 0.297*175 (ln 0.00375)

= -290.33 kPa / m3

k = -(1/v) / (-290.33)

k = -(1/0.00375) / (-290.33)

k = 0.9185 /kPa

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