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 figure. Initially spring is relaxed and 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 slide towards right to a distance $x$ . Find final pressure and temperature of the gas. (Given atmospheric pressure $= P_{0}$ and area of cross-section of the cylinder $= A$ ).

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Accepted Answer

Now, when the piston is relaxed initially, the volume and temperature of the piston is equal to $V_{0}$ and $T_{0}$ respectively. Then, the initial pressure of the gas in the cylinder should be $= P_{0}$

Let the pressure of the cylinder be $P$ when the gas is heated.

So, the free body diagram of the piston will be,

This implies that, $PA = P_{0} A + kx$

$\Rightarrow P = P_{0} + \frac{kx}{A} . . . (i)$

Since the number of moles in the cylinder must be constant.

Therefore, we can write:

$n = \frac{P_{0} V_{0}}{{RT}_{0}} = \frac{P (V_{0} + ΔV)}{RT}$

Here $ΔV = AxΔV = Ax$

$\Rightarrow T = \frac{{PT}_{0}}{P_{0} V_{0}} (V_{0} + Ax) . . . . (ii)$

From Equation. $(i)$ and $(ii)$ we get,

$T = \frac{T_{0}}{P_{0} V_{0}} (P_{0} + \frac{Kx}{A}) (V_{0} + Ax)$

Hence, the final pressure is $P = P_{0} + \frac{kx}{A}$

And, the final temperature is $T = \frac{T_{0}}{P_{0} V_{0}} (P_{0} + \frac{Kx}{A}) (V_{0} + Ax)$

B M Sharma Solutions for Chapter: Kinetic Theory of Gases, Exercise 1: ILLUSTRATIONS

B M Sharma Physics Solutions for Exercise - B M Sharma Solutions for Chapter: Kinetic Theory of Gases, Exercise 1: ILLUSTRATIONS

Questions from B M Sharma Solutions for Chapter: Kinetic Theory of Gases, Exercise 1: ILLUSTRATIONS with Hints & Solutions

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