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

Given:
Amount of heat given to the gas = Q
So, ∆Q = Q
Work done by the gas, ∆W = $\frac{\mathrm{Q}}{2}$
From the first law of thermodynamics,
∆Q = ∆W + ∆U
$\Rightarrow ∆U=\mathrm{Q}-\frac{\mathrm{Q}}{2}=\frac{\mathrm{Q}}{2}$
For a monoatomic gas,
∆U = $\frac{3}{2}$ nRdT
$\Rightarrow \frac{\mathrm{Q}}{2}=ndT\times \frac{3}{2}R$
$\Rightarrow$Q = 3nRdT

Again, for expansion at constant pressure,
Q = nC_pdT,
where C_p is the molar heat capacity at constant pressure.
So, 3RndT = nC_pdT
⇒ C_p = 3R