The volume $$V$$ of a given mass of monoatomic gas changes with temperature $$T$$ according to the relation $$V = KT^{\frac{2}{3}}$$. The workdone when temperature changes by 90 K will be $$xR$$. The value of $$x$$ is [R universal gas constant]

We are given a monoatomic gas with $$V = KT^{2/3}$$ and need to find the work done when the temperature changes by 90 K.

For 1 mole of an ideal gas, $$PV = RT$$, so $$P = \frac{RT}{V} = \frac{RT}{KT^{2/3}} = \frac{R}{K} T^{1/3}$$.

The work done is $$W = \int P \, dV$$. Since $$V = KT^{2/3}$$, we have $$dV = K \cdot \frac{2}{3} T^{-1/3} \, dT$$.

Substituting, $$W = \int \frac{R}{K} T^{1/3} \cdot K \cdot \frac{2}{3} T^{-1/3} \, dT = \int \frac{2R}{3} \, dT$$.

For a temperature change of $$\Delta T = 90$$ K, $$W = \frac{2R}{3} \times 90 = 60R$$.

Therefore, $$x = 60$$.