Question

Derive an expression for β = 1/V(∂V/∂T)P for such a gas in terms of b(T) , db(T)/dT , P , and Vm .

Express your answer in terms of the variables b(T) , db(T)/dT , P , and Vm .

Answer 1

Assume that the equation of state for a gas can be written in the form P(V_m−b(T)) = RT.

We have to derive an expression for β = 1/V(∂V/∂T)_P ......(1)

V_m=molar volume of the gas=V/n [ n=number of moles of the gas; V=volume of the gas]

V = V_mxn

Putting this in equation (1), we will get

β = 1/V_m(∂V_m/∂T)_P .......(2)

Now we have to find the value of (∂V_m/∂T)_P

First convert the equation P(V_m−b(T)) = RT in term of V_m

V_m=(RT/P)+b(T) .......(3)

Then differentiate the equation (3) with respect to (w.r.t.) T at constant P

(∂V_m/∂T)_P = (R/P)+(∂b(T)/∂T)_P

Put this expression in equation (1), we will get

β = 1/V_m[(R/P)+(∂b(T)/∂T)_P]

To remove the R value in the equation put the value of R/P=(V_m−b(T))/T to the above equation

β = 1/V_m{[(V_m−b(T))/T]+(∂b(T)/∂T)_P}