To solve this problem, we can use the first law of thermodynamics, which states:

$$\Delta Q = \Delta U + W$$

where:

• $$\Delta Q$$ is the heat transferred to the system, in this case, $$48 \mathrm{~J}$$.
• $$\Delta U$$ is the change in internal energy of the system.
• $$W$$ is the work done by the system.

For one mole of an ideal gas, the change in internal energy can be calculated using the equation:

$$\Delta U = nC_{v}\Delta T$$

where:

• $$n$$ is the number of moles, which is $$1$$ in this case.
• $$C_{v}$$ is the molar specific heat capacity at constant volume.
• $$\Delta T$$ is the change in temperature, in Kelvins.

Given that we are dealing with helium, a monatomic ideal gas, we know that:

$$C_{v} = \frac{3}{2}R$$

Substituting the given values, we get:

$$\Delta U = (1) \frac{3}{2}(8.3 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1})(2^{\circ} \mathrm{C})$$

Since a change in temperature of $$1^{\circ} \mathrm{C}$$ is equivalent to a change in temperature of $$1 \mathrm{~K}$$, we can directly substitute $$2^{\circ} \mathrm{C}$$ as a $$2 \mathrm{~K}$$ change without needing to convert Celsius to Kelvin as both scales have the same magnitude of degree.

Therefore,

$$\Delta U = 1 \times \frac{3}{2} \times 8.3 \times 2 = 24.9 \mathrm{~J}$$

Now, substituting $$\Delta U$$ and $$\Delta Q$$ into the first law of thermodynamics:

$$48 = 24.9 + W$$

Solving for $$W$$ we find:

$$W = 48 - 24.9 = 23.1 \mathrm{~J}$$

Thus, the work done by the gas is $$23.1 \mathrm{~J}$$, which corresponds to Option A.