Polytropic process has an equation $PV^{x}=k$.
But from this how can we conclude $C_v=\dfrac{R}{x-1}$ or $C_p=\dfrac{xR}{x-1} $ ?
I could'nt find any proof on the net.
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This is my attempt at a more general approach /alternative to what is presented on Polytropic Process Molar Heat Capacity
Please note attempt rather than answer, so I am trying to put it all together, but this attempt is based on notes I took only last week, but hopefully it will still be useful in some way.
I have used $x $ to avoid (hopefully), confusion with the $n$ usually used as the polytropic power, and to correspond with your notation.
The specific heat for a polytropic process is given by $$C_n = C_v\frac {\gamma − n} {1 − n}$$ where $$\gamma = \frac {C_p} {C_v} $$
$C_n$ corresponds to the value of $n$. If $n = 0$ (an isobaric process) we have $$C_0 = C_v\frac {\gamma} {1} = C_v \frac {C_p }{C_v} = C_p $$
The source of these equations above is Wikipedia Polytropic Processes
Using: $$\delta Q = n C_x d T$$ and
$$dU = n C_v dT $$
Starting with: $$dU = dQ + \delta W $$
Now, just concentrating on Work, leaving anything else for you to slog through and subsitute the relevant expressions, sorry:)
$$W = − \frac {P_i V_i} {( x − 1 )} \left[\left (\frac {V_i} {V_f }\right)^{ x − 1} − 1 \right] $$
$$= − \frac {x R T_i} {( x − 1 )} \left[\left (\frac {V_i} {V_f }\right )^{ x − 1} − 1 \right]$$
Incorporating the ideal gas law into the polytropic equation gives
$$\left (\frac { V_i }{V_f }\right) ^{x − 1} = \frac {T_f} {T_i} $$
Putting all this together results in :$$ W = − \frac {x R} {( x − 1 )} ( T_f − T_i ) $$
Substituting this into the first law gives: $$n C_x \Delta T = x C_p \Delta T −\frac { x R\Delta T}{(x − 1)} $$
Finding a common denominator for the right hand side:
$$C_p = \frac { x R}{(x − 1)}$$
