Question

Consider a gas whose equation of state is \(P(v-a)=\) \(R T,\) where \(a\) is a positive constant. Is it possible to cool this gas by throttling?

Short answer

Answer: No, it is not possible to cool this gas by throttling because the Joule-Thomson coefficient is zero. This means that the temperature of the gas remains constant during the throttling process for this particular gas.

Step-by-step solution

Express Enthalpy as a function of Temperature and Volume

Given the equation of state: $$P(v-a) = RT$$ Enthalpy is given by: $$H = U + PV = C_vT + RT(v - a)$$ where \(C_v\) is the molar heat capacity at constant volume.

Find the partial derivative of Enthalpy with respect to Temperature

Differentiating enthalpy H by temperature T, keeping the volume constant: $$\left(\frac{\partial H}{\partial T}\right)_v = C_v + R(v - a)$$

Find the partial derivative of Enthalpy with respect to Volume

Differentiating enthalpy H by volume V, keeping the temperature constant: $$\left(\frac{\partial H}{\partial v}\right)_T = RT$$

Calculate the Joule-Thomson coefficient

The Joule-Thomson coefficient is given by: $$\mu = \left(\frac{\partial T}{\partial P}\right)_H = \left(\frac{\partial T}{\partial H}\right)_v \left(\frac{\partial H}{\partial P}\right)_T$$ But, noticing that the Enthalpy is only dependent on Temperature and could not be further differentiated for Pressure, we could say that $$\left(\frac{\partial H}{\partial P}\right)_T = 0$$, hence $$\mu = \left(\frac{\partial T}{\partial P}\right)_H = 0$$

Conclusion on cooling by throttling

Since the Joule-Thomson coefficient is zero, it means that this gas cannot be cooled by throttling. The temperature of the gas remains constant during the throttling process for this particular gas, following the equation of state given.