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 final pressure of gas in $B$ to that of gas in $A$ is

Question

Verified by Toppr

Answer 1

Consider the $P - V$ diagram shown for the container $A$ and container $B$ .

Both the process involves compression of the gas

(i) Isothermal compression (Gas $A$ ) (during $1 \to 2$ )

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

$⟹ P_{0} (2 V_{0})^{γ} = P_{2} (V_{0})^{γ}$

$⟹ P_{0} (2 V_{0}) = P_{2} (V_{0})$

(II) Adiabatic compression (Gas $B$ ) (during $1 \to 2)$

$P_{1} V_{1}^{γ} = P_{2} V_{2}^{γ}$

$⟹ P_{0} (2 V_{0})^{γ} = P_{2} (V_{0})^{γ}$

$⟹ P_{2} = (\frac{2 V _{0}}{V _{0}})^{γ}$ $P_{0} = (2)^{γ} P_{0}$

Hence $\frac{( P _{2} ) _{B}}{( P _{2} ) _{A}} =$ Ratio of final pressure $= \frac{( 2 ) ^{γ} P _{0}}{2 P _{0}} = 2^{γ - 1}$

where, $γ$ is the ratio of specific heat capacities for the gas.

Answer 2

Consider the

P - V

diagram shown for the container

A

and container

B

.

Both the process involves compression of the gas

(i) Isothermal compression (Gas

A

) (during

1 \to 2

)

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

⟹ P_{0} (2 V_{0})^{γ} = P_{2} (V_{0})^{γ}

⟹ P_{0} (2 V_{0}) = P_{2} (V_{0})

(II) Adiabatic compression (Gas

B

) (during

1 \to 2)

P_{1} V_{1}^{γ} = P_{2} V_{2}^{γ}

⟹ P_{0} (2 V_{0})^{γ} = P_{2} (V_{0})^{γ}

⟹ P_{2} = (\frac{2 V _{0}}{V _{0}})^{γ}

P_{0} = (2)^{γ} P_{0}

Hence

\frac{( P _{2} ) _{B}}{( P _{2} ) _{A}} =

Ratio of final pressure

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

where,

γ

is the ratio of specific heat capacities for the gas.