Two different adiabatic paths for the same gas intersect two isothermal curves as shown in P - V diagram. The relation between the ratio and the ratio is

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Q.
Two different adiabatic paths for the same gas intersect two isothermal curves as shown in P - V diagram. The relation between the ratio $\frac{V_{a}}{V_{d}}$ and the ratio $\frac{V_{b}}{V_{c}}$ is [2024]

1 $\frac{V_{a}}{V_{d}} = \frac{V_{b}}{V_{c}}$

2 $\frac{V_{a}}{V_{d}} \neq \frac{V_{b}}{V_{c}}$

3 $\frac{V_{a}}{V_{d}} = {(\frac{V_{b}}{V_{c}})}^{2}$

4 $\frac{V_{a}}{V_{d}} = {(\frac{V_{b}}{V_{c}})}^{- 1}$

Ans.
(1)

For adiabatic process

$T V^{γ - 1} = constant$

$T_{a} \cdot V_{a}^{γ - 1} = T_{d} \cdot V^{γ - 1}$

${(\frac{V_{a}}{V_{d}})}^{γ - 1} = \frac{T_{d}}{T_{a}}$ ...(i)

$T_{b} \cdot V_{b}^{γ - 1} = T_{c} \cdot V_{c}^{γ - 1}$

${(\frac{V_{b}}{V_{c}})}^{γ - 1} = \frac{T_{c}}{T_{b}}$ ...(ii)

As, $T_{d} = T_{c}$ and $T_{a} = T_{b}$ $\Rightarrow \frac{V_{a}}{V_{d}} = \frac{V_{b}}{V_{c}}$

Solution

Q.
Two different adiabatic paths for the same gas intersect two isothermal curves as shown in P - V diagram. The relation between the ratio $\frac{V_{a}}{V_{d}}$ and the ratio $\frac{V_{b}}{V_{c}}$ is [2024]

1 $\frac{V_{a}}{V_{d}} = \frac{V_{b}}{V_{c}}$

2 $\frac{V_{a}}{V_{d}} \neq \frac{V_{b}}{V_{c}}$

3 $\frac{V_{a}}{V_{d}} = {(\frac{V_{b}}{V_{c}})}^{2}$

4 $\frac{V_{a}}{V_{d}} = {(\frac{V_{b}}{V_{c}})}^{- 1}$

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Solution

Q. Two different adiabatic paths for the same gas intersect two isothermal curves as shown in P - V diagram. The relation between the ratio VaVd and the ratio VbVc is [2024]

1 VaVd=VbVc

2 VaVd≠VbVc

3 VaVd=(VbVc)2

4 VaVd=(VbVc)-1

Ans. (1) For adiabatic process TVγ-1=constant Ta·Vaγ-1=Td·Vγ-1 (VaVd)γ-1=TdTa ...(i) Tb·Vbγ-1=Tc·Vcγ-1 (VbVc)γ-1=TcTb ...(ii) As, Td=Tc and Ta=Tb⇒VaVd=VbVc

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Q.
Two different adiabatic paths for the same gas intersect two isothermal curves as shown in P - V diagram. The relation between the ratio $\frac{V_{a}}{V_{d}}$ and the ratio $\frac{V_{b}}{V_{c}}$ is [2024]

1 $\frac{V_{a}}{V_{d}} = \frac{V_{b}}{V_{c}}$

2 $\frac{V_{a}}{V_{d}} \neq \frac{V_{b}}{V_{c}}$

3 $\frac{V_{a}}{V_{d}} = {(\frac{V_{b}}{V_{c}})}^{2}$

4 $\frac{V_{a}}{V_{d}} = {(\frac{V_{b}}{V_{c}})}^{- 1}$