Ncert Exercises & Solutions

Q. 5 Consider two containers A and B containing identical gases at the same pressure, volume and temperature. The gas in container A is compressed to half of its original volume isothermally while the gas in container B is compressed to half of its original value adiabatically. The ratio of the final pressure of gas in B to that of gas in A is:

a. $2^{γ - 1}$

b. ${(\frac{1}{2})}^{γ - 1}$

c. ${(\frac{1}{1 - γ})}^{2}$

d. ${(\frac{1}{γ - 1})}^{2}$

(1) Hint: Use the conditions for the isothermal process and adiabatic process.

Consider the p-V diagrams shown for container A (isothermal) and for container B (adiabatic).

Both the processes involve compression of the gas.

Step 1: Find the final pressure for gas in A.

For isothermal process (gas A) $(d u r i n g 1 \to 2)$ :

$P_{1} V_{1} = P_{2} V_{2}$

$\Rightarrow$ $P_{0} (2 V_{0}) = P_{2} (V_{0})$

$\Rightarrow$ $P_{2} = 2 P_{0}$

Step 2: Find the final pressure for gas in B.

For the adiabatic process, (gas B) (during 1

\to

2):

P_{1} {V_{1}}^{γ} = P_{2} {V_{2}}^{γ}

\Rightarrow P_{0} {(2 V_{0})}^{γ} = P_{2} {(V_{0})}^{γ}

\Rightarrow P_{2} = {(\frac{2 V_{0}}{V_{0}})}^{γ} P_{0} = {(2)}^{γ} P_{0}

Hence, the ratio of final pressures =

\frac{{(2)}^{γ} P_{0}}{2 P_{0}} = 2^{γ - 1}

where

γ

is the ratio of specific heat capacities for the gas.

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