Two thermodynamical processes are shown in the figure. The molar heat capacity for process $$A$$ and $$B$$ are $$C_A$$ and $$C_B$$. The molar heat capacity at constant pressure and constant volume are represented by $$C_P$$ and $$C_V$$, respectively. Choose the correct statement.

In the graph, the lines have positive slope in the (log⁡V,log⁡P) plane, so they represent

$$\log P=n\log V$$

or

$$P=V^n$$

which can be written as

This is a polytropic process with index

m=−n

For a polytropic process, molar heat capacity is

Substituting m=−n,

$$C=C_V+\frac{R}{1+n}$$

For process A, slope is given by

$$\tan⁡θ=γ$$

so

$$n=γ$$

Hence

$$C_A=C_V+\frac{R}{1+\gamma}$$

For process BBB, angle is $$45^{\circ}$$, so

n=1

and

$$C_B=C_V+\frac{R}{2}$$

Also,

$$C_P=C_V+R$$

Now comparing:

$$C_P>C_B$$

because

$$R>\frac{R}{2}$$

Also since γ>1,

$$\frac{R}{2}>\frac{R}{1+\gamma}$$

so

$$C_B>C_A$$

and clearly

$$C_A>C_V$$

Therefore,

$$CP>CB>CA>CV$$