$$0.08$$ kg air is heated at constant volume through $$5°C$$. The specific heat of air at constant volume is $$0.17 \text{ kcal kg}^{-1} \text{ °C}^{-1}$$ and $$1 \text{ J} = 4.18 \text{ joule cal}^{-1}$$. The change in its internal energy is approximately.

For heating at constant volume, the change in internal energy equals the heat added:

$$\Delta U = m c_v \Delta T$$

Given: $$m = 0.08$$ kg, $$c_v = 0.17$$ kcal kg$$^{-1}$$ °C$$^{-1}$$, $$\Delta T = 5°C$$.

$$\Delta U = 0.08 \times 0.17 \times 5 = 0.068 \text{ kcal}$$

Converting to joules ($$1 \text{ cal} = 4.18 \text{ J}$$, so $$1 \text{ kcal} = 4180 \text{ J}$$):

$$\Delta U = 0.068 \times 4180 = 284.24 \text{ J} \approx 284 \text{ J}$$

The answer is approximately $$284$$ J, which corresponds to Option (3).