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Step-by-Step Solution
Step 1: Express the given pressure-volume relationship
The problem states that the pressure of the gas depends on its volume according to
P = k \, V^{3}.
We can rewrite this as
P \, V^{-3} = k,
which indicates a polytropic process of the form
P V^{x} = \text{constant},
with
x = 3.
However, note that we often write the exponent in the form
V^{-3}
meaning
x = -3
for the standard polytropic form
P \, V^{x} = \text{constant}.
Step 2: Identify the polytropic index
From
P = k\,V^{3},
we rearrange to
P\,V^{-3} = k.
By comparing with
P\,V^{x} = \text{constant},
we see
x = -3.
Step 3: Use the formula for work done in a polytropic process
For an ideal gas undergoing a polytropic process with polytropic index
x,
the work done
W
when the temperature changes from
T_{1}
to
T_{2}
can be written (for this specific derivation) as:
W = \frac{n R (T_{1} - T_{2})}{x - 1}.
In our case,
x = -3.
Step 4: Substitute the initial and final temperatures
The temperature changes from
100^\circ \mathrm{C}
to
300^\circ \mathrm{C}.
Recall that we can use these temperatures in the relation for an ideal gas. For the net change in internal energy or the temperature interval, we can treat the difference
(T_1 - T_2)
in degrees Kelvin the same as the difference in degrees Celsius, since only the difference matters for the formula. Hence,
T_1 = 100 + 273 = 373 \text{ K}, \quad T_2 = 300 + 273 = 573 \text{ K}.
But for computing the difference in the formula
\,T_1 - T_2\,
in
^\circ\mathrm{C}
or
\mathrm{K}
will yield the same numerical difference. So we can also directly use
100 - 300 = -200
\text{ (as a difference)}.
Step 5: Calculate the work done
Substitute
x = -3,
and
(T_{1} - T_{2}) = (100^\circ \mathrm{C} - 300^\circ \mathrm{C}) = -200
into the formula:
W = \frac{n R\,(T_{1} - T_{2})}{x - 1}
= \frac{n R \times (-200)}{-3 - 1}
= \frac{-200\,n R}{-4}
= 50\,n R.
Step 6: State the final answer
Therefore, the work done is
50 \, n R.
