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If the equation of state for 1 mole of a gas is (p+(a)/(V^(2)))V=RT prove that p is a state function and hence dpis an exact differential.

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SOLUTION :`DP` would be an exact differential if `(del^(2)p)/(delVdelV)` as `p=f(V,T)`
Given that `(p+(a)/(V^(2)))=RT`
or `p=(RT).(V)=(a)/(V^(2))`…………….(1)
DIFFERENTIATING w.R.t V at constant T, we get
`((del p)/(del V))_(T)=-(RT)/(V^(2))+(2a)/(V^(3))`
Differentiating w.r.t T at constant V, we get
`(del^(2) p)/(del T del V)=-(R )/(V^(2))`............(2)
Again differenitiating Eqn. (1) first w.r.t T at constant V and then w.r.t V at constant T, we get
`(del^(2))/(del V del T)=-(R )/(V^(2))`...........(3)
From Eqns. (2) and (3) we have
`(del^(2)p)/(del Tdel V)=(del^(2)p)/(del V del T)`
Thus dp is an exact differential and p is a state function.


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