# Why does this thermodynamics problem seem to have two answers? [closed]

The problem goes as: For an ideal gas, molar heat capacity varies as: $$C = C_v + aV$$, where $$a$$ is a constant. Now we are asked to find a relation between Temperature and Volume.

The way I approached the problem is by comparing the general expression for the heat capacity of an ideal gas to this one. The general expression is: $$C = C_v +\frac{R}{1-n}$$ where $$n$$ is the coefficient in the general polytropic process. By comparing the two equations, we can say that:
$$\frac{R}{1-n} = aV$$
Also, we know that $$TV^{n-1} = k$$, where $$k$$ is a constant. So we can just replace $$n-1$$ in the second equation get a relation as: $$TV^{-\frac{R}{aV}} = k$$. But that is not the answer, and that is what I cannot understand. What went wrong here?

The method to get the 'correct' answer is:
$$TV^{n-1}=k$$
$$\Rightarrow$$ $$T(n-1)dV + VdT = 0$$

$$\Rightarrow$$ $$(1-n)=\frac{VdT}{TdV} = \frac{R}{aV}$$

$$\Rightarrow \int \frac{dT}{T} = \int\frac{RdV}{aV^2}$$

$$\Rightarrow ln(T) = -\frac{R}{aV} + k'$$

$$\Rightarrow T=e^{-\frac{R}{aV}+k'}$$
This is more of a mathematical question, But I couldn't boil down this thermodynamics problem into a math problem, and I guess there is a chance of physics being involved here and there.
So why is the second answer correct? And what is wrong in the first approach?

If you go through this SE post, $$C=C_V+P\frac{dV}{dT}$$ where we have taken the number of moles to be one. Comparing with the expression given $$P\frac{dV}{dT}=aV\Rightarrow \frac{1}{aV^2}dV=\frac{1}{RT}dT$$ Integratin leads to
$$T=T_0e^{-R/aV}$$ As required. The way you have done is valid only for polytropic process no in general.
A Polytropic process is the one which follows: $$pV^n=C$$ which is not general. A general process for an ideal gas defined as $$p=f(V)$$ For example, the process might be like $$p=V^3+V^2 e^{-\lambda V}$$ which is not polytropic!