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Pressure

An ideal gas ...

Question

An ideal gas is filled in an insulated cylinder fitted with a movable piston connected with an ideal spring (of force constant $K$ ) as shown in the figure. Initially, the spring is relaxed and the gas in the piston has volume $V_{0}$ and temperature $T_{0}$ . Now the gas in the cylinder is heated slowly so that the piston slides towards right to a distance $x$ . Find the final pressure and temperature of the gas. (Given atmospheric pressure = $P_{0}$ and area of cross-section of the cylinder is $A$ ).

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Solution

From the figure after heating we can say

$P_{g a s} A = k x + P_{0} A$

$P_{f} = P_{g a s} = \frac{k x}{A} + P_{0}$

Also we know numbe rof moles remain
constant. So
$\begin{matrix} n_{i} & = n_{f} \frac{P_{i} V_{i}}{R T_{i}} & = \frac{P_{i} v_{f}}{R T_{f}} \end{matrix}$

$\begin{matrix} \frac{P_{0} V_{0}}{T_{0}} & = \frac{(P_{0} + \frac{k x}{A}) (V_{0} + x A)}{T_{f}} T_{f} & = (P_{0} + \frac{k x}{A}) \frac{(V_{0} + x A) T_{0}}{P_{0} v_{0}} \end{matrix}$

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