$1 mole$ of an ideal monoatomic gas is confined in a cylinder fitted with a piston as shown in figure. Take mass of the piston is $0.5 K g,$ spring constant, $K = 1000 N / m$ , atmospheric pressure $P_{0} = 10^{5} N / m^{2},$ CSA of piston $A = 50 {cm}^{2}$ If energy of $50 J$ has been supplied to system then piston moves up by $5 c m$ [take $g = 10 m / s^{2})$ Initially spring is in its natural position. Mark the correct statement(s).

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initial pressure of the gas is $1.01 \times 10^{5} N / m^{2}$

increase in internal energy of the system is $24.625 J$

for mentioned change, the temperature of gas decreases by $2 k$

for mentioned change, the temperature of gas increases by $2 k$

answer is A, B, C.

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Detailed Solution

Let initial pressure of the gas is P, then

$\begin{array}{l} p A = m g + P_{0} A \\ \Rightarrow P = P_{0} + \frac{m g}{A} = 10^{5} + 0.01 \times 10^{5} = 1.01 \times 10^{5} N / m^{2} \end{array}$

Work done by gas, $W = P_{0} d V + m g x + \frac{k x^{2}}{2}$

$\begin{array}{l} W = 10^{5} + 50 \times 10^{- 4} \times 0.05 + 5 \times 0.05 + \frac{1000 \times {0.05}^{2}}{2} \\ = 25 + 0.25 + 0.125 = 25.375 J \\ From 1 st law, dQ = dU + W \\ \Rightarrow dU = 50 - 25.375 = 24.625 J \end{array}$