The amount of heat needed to raise the temperature of 4 moles of a rigid diatomic gas from 0 $$^\circ$$C to 50 $$^\circ$$C when no work is done is ($$R$$ is the universal gas constant)

A rigid diatomic gas molecule has 5 degrees of freedom: 3 translational and 2 rotational. By the equipartition theorem, the molar heat capacity at constant volume is $$C_V = \frac{f}{2}R = \frac{5}{2}R$$, where $$R$$ is the universal gas constant.

Since no work is done by the gas (constant volume process), the first law of thermodynamics gives $$Q = \Delta U = nC_V\Delta T$$.

With $$n = 4$$ moles and $$\Delta T = 50 - 0 = 50$$ K:

$$Q = 4 \times \frac{5}{2}R \times 50 = 4 \times 2.5 \times 50 \times R = 500R$$

The heat required is $$500R$$.