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Step-by-Step Solution
Step 1: Identify the given process
The problem states that the gas follows the relation
$PV^{\frac{3}{2}} = K,$
where $K$ is a constant. This indicates how pressure $P$ and volume $V$ change during the given process.
Step 2: Write down the expression for work done
For a quasi-static process, the work done by the gas in expanding (or compressing) from volume $V_1$ to $V_2$ is given by the definite integral:
$W = \int_{V_1}^{V_2} P \, dV.$
Step 3: Express $P$ in terms of $V$
From the relation
$PV^{\frac{3}{2}} = K,$
we solve for $P$:
$P = \frac{K}{V^{\frac{3}{2}}}.$
Step 4: Substitute $P$ into the work integral
Substitute
$P = \frac{K}{V^{\frac{3}{2}}}$
into
$W = \int_{V_1}^{V_2} P \, dV:$
$W = \int_{V_1}^{V_2} \frac{K}{V^{\frac{3}{2}}} \, dV
= K \int_{V_1}^{V_2} V^{-\frac{3}{2}} \, dV.$
Step 5: Evaluate the integral
We know that
$\int V^{n} \, dV = \frac{V^{n+1}}{n+1}$
(for $n \neq -1$). Here, $n = -\frac{3}{2}$, so:
$\int V^{-\frac{3}{2}} \, dV = -2 \, V^{-\frac{1}{2}}.$
Hence,
$W = K \left[ -2 \, V^{-\frac{1}{2}} \right]_{V_1}^{V_2}
= K \left( -2\,V_2^{-\frac{1}{2}} + 2\,V_1^{-\frac{1}{2}} \right).$
Step 6: Express $K$ in terms of initial or final states
From
$PV^{\frac{3}{2}} = K,$
we can write
$K = P_1 \, V_1^{\frac{3}{2}} = P_2 \, V_2^{\frac{3}{2}}.$
Let us use
$K = P_1 \, V_1^{\frac{3}{2}}.$
Thus,
$W
= P_1 \, V_1^{\frac{3}{2}} \left( -2\,V_2^{-\frac{1}{2}} + 2\,V_1^{-\frac{1}{2}} \right)
= -2 P_1 V_1^{\frac{3}{2}} \frac{1}{\sqrt{V_2}} + 2 P_1 V_1^{\frac{3}{2}} \frac{1}{\sqrt{V_1}}.
Notice that
$V_1^{\frac{3}{2}} \div \sqrt{V_1} = V_1,$
so the expression simplifies. By similarly relating $P_2, V_2$ for consistency, the final simplified result for the work is:
$W = 2 \bigl( P_1 V_1 - P_2 V_2 \bigr).$
Step 7: State the final answer
Thus, the work done when the gas changes its state from
$(P_1, V_1)$
to
$(P_2, V_2)$
under the given process
$PV^{\frac{3}{2}} = \mathrm{constant}$
is:
$W = 2 \bigl( P_1 V_1 - P_2 V_2 \bigr).$
Hence, the correct choice is
$2\left(P_1 \, V_1 - P_2 \, V_2\right).$
