$$7$$ mole of certain monoatomic ideal gas undergoes a temperature increase of $$40 \text{ K}$$ at constant pressure. The increase in the internal energy of the gas in this process is (Given $$R = 8.3 \text{ J K}^{-1} \text{ mol}^{-1}$$)

We need to find the increase in internal energy when $$7$$ moles of a monoatomic ideal gas undergoes a temperature increase of $$\Delta T = 40 \text{ K}$$ at constant pressure.

For an ideal gas, the change in internal energy depends only on temperature change (not on the process). For a monoatomic ideal gas:

$$\Delta U = n C_V \Delta T$$

where $$C_V = \dfrac{3}{2}R$$ for a monoatomic ideal gas.

$$\Delta U = n \times \frac{3}{2}R \times \Delta T$$

$$\Delta U = 7 \times \frac{3}{2} \times 8.3 \times 40$$

$$\Delta U = 7 \times 1.5 \times 8.3 \times 40$$

$$\Delta U = 7 \times 1.5 \times 332$$

$$\Delta U = 7 \times 498$$

$$\Delta U = 3486 \text{ J}$$

The correct answer is Option B: $$3486 \text{ J}$$.