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 process described is a polytropic process of the form $PV^n = \text{constant}$, where the polytropic index $n = 3$.

The molar heat capacity ($C$) of an ideal gas in a polytropic process is given by the formula:
$$ C = C_V + \frac{R}{1-n} $$

For a monatomic ideal gas, the molar heat capacity at constant volume is:
$$ C_V = \frac{3}{2}R $$

Substituting the values into the formula:
$$ C = \frac{3}{2}R + \frac{R}{1-3} $$
$$ C = \frac{3}{2}R + \frac{R}{-2} $$
$$ C = \frac{3}{2}R - \frac{1}{2}R $$
$$ C = \frac{2}{2}R = R $$

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