n moles of an ideal monoatomic gas undergoes a process from $1 \to 4$ as shown in the figure. $T_{1} = 500 K, T_{4} = 200 K, P_{1} = 10^{5}$ Pa and $P_{2} = 4 \times 10^{5} P a .$ In state 3, $3 V_{3} = V_{1} .$ The change in internal energy from state 3 to state 4 is x Joules. The value of ‘x’ is (Take no. of moles $n = 1 / R,$ where R = gas constant)

Moderate

Solution

$\begin{array}{l} \frac{P - 4 \times 10^{5}}{T - 500} = \frac{- 3 \times 10^{5}}{- 300} \\ \Rightarrow P = 10^{3} T - 10^{5} \\ \Rightarrow \frac{nRT}{V} = 10^{3} T - 10^{5} \end{array}$
$\begin{array}{l} A s n = \frac{1}{R} \\ \Rightarrow \frac{1}{V} = 10^{3} - \frac{10^{5}}{T} [s t r a i g h t l i n e f r o m 2 t o 4] \\ A t p o i n t 1, V_{1} = \frac{n R T_{1}}{P_{1}} = \frac{500}{10^{5}} m^{3} \\ G i v e n : 3 V_{3} = V_{1} \\ \Rightarrow V_{3} = \frac{V_{1}}{3} = \frac{500}{3 \times 10^{5}} m^{3} \\ A t p o i n t 3, \frac{1}{V_{3}} = 10^{3} - \frac{10^{5}}{T_{3}} \\ \Rightarrow \frac{3 \times 10^{5}}{500} = 1000 - \frac{10^{5}}{T_{3}} \\ \Rightarrow 600 = 1000 - \frac{10^{5}}{T_{3}} \\ \Rightarrow T_{3} = 250 K \\ C h a n g e i n i n t e r n a l e n e r g y f r o m s t a t e 3 t o s t a t e 4, \\ △ U = n C_{V} (T_{4} - T_{3}) = \frac{3}{2} \times (200 - 250) = - 75 J \end{array}$

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