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arpon

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## Homework Statement

Show that for a gas obeying the van der Waals equation ##\left(P+\frac{a}{v^2}\right)(v-b)=RT##, with ##c_v## a function of ##T## only, an equation for an adiabatic process is $$T(v-b)^{R/c_v}=constant$$

## Homework Equations

##TdS=c_vdT+T\left(\frac{\partial P}{\partial T}\right)_v dv##

## The Attempt at a Solution

For reversible adiabatic process, ##dS=0##.

So, from the third ##TdS## equation,

$$c_vdT+T\left(\frac{\partial P}{\partial T}\right)_v dv=0$$

$$c_vdT+T\left(\frac{R}{v-b}\right)dv=0~~~$$ [Using equation of state]

$$c_v\frac{dT}{T}=-\frac{RdV}{v-b}$$

If ##c_v## is a constant, integrating both sides, we have:

$$T(v-b)^{R/c_v}=constant$$

But, in this case, ##c_v## is a function of ##T##.

Any help would be appreciated.