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  • An ideal gas is taken from the state A (pressure P, volume V) to the state B (pressure P/2, volume 2V) along a straight line path in the P-V diagram. Select the correct statements from the followin

An ideal gas is taken from the state A (pressure P, volume V) to the state B (pressure P/2, volume 2V) along a straight line path in the P-V diagram. Select the correct statements from the following :

Option: 1

The work done by the gas in the process A to B exceeds the work that would be done by it if the system were taken from A to B along an isotherm


Option: 2

In the T-V diagram, the path AB becomes a part of a parabola


Option: 3

In going from A to B, the temperature T of the gas first increases to a maximum value and then decreases


Option: 4

All of the above


Answers (1)

best_answer

(a) Work done = Area under P-V graph


\therefore \mathrm{W}_{\text {given process }}                   >\mathrm{W}_{\text {isothermal process }}

(b) In the given process P-V equation will be of a straight line with negative slope and positive intercept i.e.,
    \mathrm{P}=-\alpha \mathrm{V}+\beta$ (Here $\alpha$ and $\beta \text{are positive constant})

\Rightarrow \quad \mathrm{PV}=-\alpha \mathrm{V}^{2}+\beta \mathrm{V}
\Rightarrow \quad \mathrm{nRT}=-\alpha \mathrm{V}^{2}+\beta \mathrm{V}
\Rightarrow \quad \mathrm{T}=\frac{1}{n R}\left(-\alpha \mathrm{V}^{2}+\beta \mathrm{V}\right)\quad \ldots(1)

This is an equation of parabola in T and V.
\mathrm{(c) \frac{d T}{d V}=0=\beta-2 \alpha \mathrm{V}}
\mathrm{\Rightarrow \quad \mathrm{V}=\frac{B}{2 \alpha}}
\mathrm{Now, \frac{d^{2} T}{d V^{2}}=-2 \alpha=-\mathrm{ve}}

i.e., \mathrm{\mathrm{T}} has some maximum value.
\mathrm{\text { Now, } \quad \mathrm{T} \propto \mathrm{PV}}
\mathrm{And (\mathrm{PA})_{\mathrm{A}}=(\mathrm{PV})_{\mathrm{B}}}
\mathrm{\Rightarrow \quad \mathrm{T}_{\mathrm{A}}=\mathrm{T}_{\mathrm{B}}}

We conclude that temperatures are same at A and B and temperature has a maximum value. Therefore, in going from A to B, T will first increase to a maximum value and then decrease.

 

 

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Rishi

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