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An ideal gas is taken from the state A (...

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 statement(s) from the following.

A

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 the isotherm.

B

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

C

In the P-T diagram, the path AB becomes a part of a hyperbola.

D

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

Text Solution

Verified by Experts

The correct Answer is:
a, b, d
The area under isotherm is smaller than area under straight line process.

In the given process PV equation will be of a straight line with negative slope and positive intercept , i.e.,
` P = - alpha V + beta (" here " alpha " and " beta " are positive constants ")`
`rArr PV = -alpha V^(2) + beta V`
`rArr nRT = - alpha V^(2) + beta V`
` rArr T = 1/(mR) (- alpha V^(2) + beta V)`
This is an equation of parabola in T and V.
Again, `(dT)/(dV) = 0 - beta - 2 alpha V rArr V = beta/(2 alpha ) `
` (d^(2) T)/(dV^(2)) = - 2 alpha = - ve , i.e.,` T has maximum value.
Now , `T propto PV " and " (PV)_(A) = (PV)_(B) rArr T_(A) = T_(B) `
We can conclude that temperatures are same at A and B and in between temperature has maximum value.
Therefore in going from A to B , T will first increase to a maxium value and then decreases.
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