A monoatomic gas performs a work of $$\frac{Q}{4}$$ where $$Q$$ is the heat supplied to it. The molar heat capacity of the gas will be ______ $$R$$.
Where $$R$$ is the gas constant.

A monoatomic gas performs work $$W = \frac{Q}{4}$$ where $$Q$$ is the heat supplied. We need to find the molar heat capacity in terms of $$R$$.

Since by the first law of thermodynamics $$Q = \Delta U + W$$ and $$W = \frac{Q}{4}$$, we have

$$Q = \Delta U + \frac{Q}{4}$$

This gives

$$Q - \frac{Q}{4} = \Delta U$$

so that

$$\frac{3Q}{4} = \Delta U$$

and hence

$$Q = \frac{4}{3}\Delta U$$

Now, for a monoatomic ideal gas with $$n$$ moles, the change in internal energy can be expressed as

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

Substituting this into the expression for $$Q$$ yields

$$Q = \frac{4}{3} \cdot n \cdot \frac{3R}{2} \cdot \Delta T = n \cdot 2R \cdot \Delta T$$

From the definition $$Q = nC\Delta T$$, it follows that

$$C = 2R$$

Therefore, the molar heat capacity of the gas is 2R.