Question

One mole of an ideal gas with adiabatic exponent

γ

whose pressure changes with volume as

P = α V

, where

α

is a constant, is expanded so that its volume increase

η

times. Find the change in internal energy and heat capacity of the gas. Initial volume of gas is

V_{0}

.

Accepted Answer

Given process: $P = α V$ $. . . (i)$

Rearranging the above equation we can write

$P V^{- 1} = α$ ( $a$ constant) $. . . (ii)$

Equation $(ii)$ shows a polytropic process with the value of polytropic constant $N = - 1$ . Thus, the molar heat capacity of gas in this process can be given as

$C = \frac{R}{γ - 1} + \frac{R}{1 - N} = \frac{R}{γ - 1} + \frac{R}{2} = \frac{R}{2} [\frac{γ + 1}{γ - 1}]$

During the process, the temperature of gas changes from $T_{1}$ to $T_{2}$ then change in internal energy of gas;

$Δ U = C_{V} Δ T = \frac{R}{γ - 1} (T_{2} - T_{1})$ $. . . (iii)$

As $n = 1$ mole and $C_{V} = \frac{R}{γ - 1}$
As initial and final volumes of the gas are $V_{0}$ and $η V_{0}$ . Thus the respective pressures in initial and final states are: $P = α V_{0}$ and $P = α η V_{0}$

From gas law we can find initial temperatures as

$T_{1} = \frac{P_{1} V_{1}}{R} = \frac{α V_{0} \cdot V_{0}}{R}$ and $T_{2} = \frac{P_{2} V_{2}}{R} = \frac{α η V_{0} \cdot η V_{0}}{R}$

Now from the equation $(iii)$ change in internal energy of the gas is given as

$Δ U = \frac{R}{γ - 1} [\frac{α η^{2} V_{0}^{2}}{R} - \frac{α V_{0}^{2}}{R}] = \frac{α V_{0}^{2}}{γ - 1} [η^{2} - 1]$

Hence, the change in internal energy is $Δ U = \frac{α V_{0}^{2}}{γ - 1} [η^{2} - 1]$

B M Sharma Solutions for Chapter: Thermodynamics, Exercise 1: ILLUSTRATIONS

B M Sharma Physics Solutions for Exercise - B M Sharma Solutions for Chapter: Thermodynamics, Exercise 1: ILLUSTRATIONS

Questions from B M Sharma Solutions for Chapter: Thermodynamics, Exercise 1: ILLUSTRATIONS with Hints & Solutions

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