Question

Three samples A, B and C of the same gas (γ = 1.5) have equal volumes and temperatures. The volume of each sample is doubled, the process being isothermal for A, adiabatic for B and isobaric for C. If the final pressures are equal for the three samples, find the ratio of the initial pressures.

Answer 1

The ratio of the initial pressures for three gas samples A, B, and C is $\mathrm{P}_{\mathrm{A}}: \mathrm{P}_{\mathrm{B}}: \mathrm{P}_{\mathrm{C}}=2: 2^{1.5}: 1$

Explanation:

Given Data

A , B and C are three sample gases , Where the process in A is Isothermal, Adiabatic in B, Isobaric in C.

Volume of each sample is doubled and they have equal final pressure for the three samples.

It is initially given that,

$V_{A}=V_{B}=V_{C} \quad \text { and } \quad T_{A}=T_{B}=T_{C}$

For A, the process is isothermic and PV = constant in case of  isothermal process.

$P_{A} V_{A}=P_{A}^{\prime} 2 V_{A}$

$P_{A}^{\prime}=P_{A} \times \frac{1}{2}$

For B, that is adiabatic operation.

$P_{B} V_{B}=P_{B}^{\prime}\left(2 V_{B}\right)^{\gamma}$

$P_{B}^{\prime}=\frac{P_{B}}{2^{1.5}}$

For C, the process is isobaric, meaning the pressure remains constant.  

So, use the same equation for gas

$P=\frac{\mathrm{nRT}}{\mathrm{V}}$

$\frac{V_{C}}{T_{C}}=\frac{V^{\prime} c}{T^{\prime} c}$

$\frac{\mathrm{V}_{\mathrm{C}}}{\mathrm{T}_{\mathrm{C}}}=\frac{2 \mathrm{V}_{\mathrm{C}}}{\mathrm{T}_{\mathrm{C}}^{\prime}}$

$\mathrm{T}_{\mathrm{c}}^{\prime}=2 \mathrm{T}_{\mathrm{C}}$

Since the final pressures are similar,

$\frac{P_{A}}{2}=\frac{P_{B}}{2^{1.5}}=P_{C}$

$\mathrm{P}_{\mathrm{A}}: \mathrm{P}_{\mathrm{B}}: \mathrm{P}_{\mathrm{C}}=2: 2^{1.5}: 1$

Therefore the ratio of the initial pressures when the final pressures are equal for the three gas samples A (isothermal), B (Adiabatic) and C (Isobaric)  $\mathrm{P}_{\mathrm{A}}: \mathrm{P}_{\mathrm{B}}: \mathrm{P}_{\mathrm{C}}=2: 2^{1.5}: 1$ (where γ = 1.5 )