First slide

Thermodynamic processes

Question

One mole of an ideal monoatomic gas is taken through process AB given by $P = α - {βV}^{2}$ (where $α$ and $β$ are positive constants) on P-V diagram. Which of the following is the correct statement regarding the given process?

Moderate

A

Temperature is maximum at $V = \sqrt{\frac{α}{3 β}}$
B

Temperature is minimum at $V = \sqrt{\frac{α}{3 β}}$
C

At point A rate of increase of temperature of gas with volume is more than at point B
D

Rate of increase of temperature of gas with volume is maximum at $V = \sqrt{\frac{α}{β}}$

Solution

$\begin{array}{l} PV = 1 \cdot RT \\ (α - {βV}^{2}) V = RT \\ T = \frac{αV}{R} - \frac{{βV}^{3}}{R} \\ \frac{dT}{dV} = \frac{α}{R} - \frac{3 {βV}^{2}}{R} \end{array}$
For maximum value of T
$\begin{array}{l} \frac{dT}{dV} = 0 \\ \Rightarrow V = \sqrt{\frac{α}{3 β}} \end{array}$

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