First slide

Gaseous state

Question

One mole of a gas changed from its initial state (15L.,2 atm) to final state (4L, 10 atm) reversibly. If this change can be represented by a straight line in P -V curve, maximum temperature (approximate) the gas attained is x × 10²K. Then find the value of x.

Moderate

Solution

Equation of line
$\begin{aligned} P - 2 = \frac{10 - 2}{4 - 15} (V - 15) \\ P - 2 = - \frac{8}{11} (V - 15) \\ P = 2 - \frac{8 V}{11} + \frac{15 \times 8}{11} \\ P = (\frac{142}{11} - \frac{8 V}{11}) \end{aligned}$
$\begin{array}{l} f (T) = \frac{1}{nR} (\frac{142 V}{11} - \frac{8 V^{2}}{11}) \\ \frac{d (F (T))}{dV} = \frac{1}{nR} (\frac{142}{11} - \frac{16 V}{11}) = 0 \\ V = \frac{142}{16} = 8 .875 \\ P = \frac{142}{11} - [\frac{8}{11} \times 8.875] = \frac{71}{11} \\ T_{Max} = \frac{PV}{nR} = \frac{71}{11} \times \frac{142}{16 \times 0 .0821} = 700 \\ = 7 \times 10^{2} K \end{array}$

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