One mole of an ideal monatomic gas undergoes a process described by the equation $PV^3 = ext{constant}$. The

Question

One mole of an ideal monatomic gas undergoes a process described by the equation $PV^3 = 	ext{constant}$. The heat capacity of the gas during this process is:

Accepted Answer

The molar heat capacity ($C$) of an ideal gas undergoing a polytropic process defined by $PV^n = 	ext{constant}$ is given by the relation derived from the First Law of Thermodynamics ($q = \Delta U - w$): 
$C = C_V + \frac{R}{1-n}$.

1. **Identify the gas type:** The gas is monatomic. From the provided data, the molar heat capacity at constant volume for monatomic gases (like Helium, Argon) is approximately $12.5 	ext{ J mol}^{-1} 	ext{ K}^{-1}$, which corresponds to $C_V = \frac{3}{2}R$ . 
2. **Identify the process:** The given equation is $PV^3 = 	ext{constant}$, so the polytropic index $n = 3$. 
3. **Calculate C:** 
$C = \frac{3}{2}R + \frac{R}{1-3}$ 
$C = \frac{3}{2}R + \frac{R}{-2}$ 
$C = \frac{3}{2}R - \frac{1}{2}R = \frac{2}{2}R = R$.

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