Question

An amount Q of heat is added to a monatomic ideal gas in a process in which the gas performs a work Q/2 on its surrounding. Find the molar heat capacity for the process.

Answer 1

The molar heat capacity for the process is $C_{p}=3 R$.

Explanation:

Step 1:

Given data in the question  

The total of heat the gas receives = Q

So, ∆Q = Q

gas Work done, ∆W = $\frac{Q}{2}$

Step 2:

From the law of thermodynamics first,

∆Q = ∆W + ∆U

$\Delta U=Q-\frac{Q}{2}=\frac{Q}{2}$

For Gas Mono atomic,

$\begin{aligned}&\Delta U=32 n R d T\\&\frac{Q}{2}=n d T \times \frac{3}{2} R\\&Q=3 n R d T\end{aligned}$

Step 3:

Once, at constant pressure for expansion,

$Q=n C_{p} d T$

Where $C_p$ is the potential for molar heat at constant pressure.

So,  

$\begin{aligned}&3 R n d T=n C p d T\\&C_{p}=3 R\end{aligned}$