1. Given process

The gas follows $PV^{\frac{3}{2}}=K$ where $K$ is constant.

So, $P=KV^{-\frac{3}{2}}$

2. Expression for work done

Work done by the gas from $A(V_1,P_1)$ to $B(V_2,P_2)$ is $W=\int_{V_1}^{V_2} P\,dV$

Substitute $P=KV^{-3/2}$: $W=\int_{V_1}^{V_2} KV^{-3/2}\,dV$

3. Integrate

$\int V^{-3/2}dV = -2V^{-1/2}$

Hence, $W=K\left[-2V^{-1/2}\right]_{V_1}^{V_2}$ $W=-2K\left(\frac{1}{\sqrt{V_2}}-\frac{1}{\sqrt{V_1}}\right)$ $W=2K\left(\frac{1}{\sqrt{V_1}}-\frac{1}{\sqrt{V_2}}\right)$

4. Rewrite in terms of $P$ and $V$

Since $K=PV^{3/2}$ we get $\frac{K}{\sqrt{V}}=PV$

So, $\frac{K}{\sqrt{V_1}}=P_1V_1, \qquad \frac{K}{\sqrt{V_2}}=P_2V_2$

Therefore, $W=2(P_1V_1-P_2V_2)$

5. Match with options

This corresponds to: $\boxed{2(P_1V_1-P_2V_2)}$ which is Option D.