Three moles of an ideal gas being initially at a temperature `T_i=273K` were isothermally expanded 5 times its initial volume and then isochorically heated so that the pressure in the final state becomes equal to that in the initial state. The total heat supplied in the process is 80kJ. Find `gamma(=(C_p)/(C_V))` of the gas.

Explanation

In the isothermal process, heat transfer to the gas is given by
`Q_1 = v RT_0 1n (V_2)/(V_1) = v RT_0 1 n eta (For eta = (V_2)/(V_1) = (p_1)/(p_2))`
In the isochoric process, `A = 0`
Thus heat transfer to the gas is given by
`Q_2 = Delta U = v C_V Delta T = (v R)/(gamma - 1) Delta T(for C_V = (R)/(gamma - 1))`
But `(p_2)/(p_1) = (T_0)/(T)`, or `T = T_0 (p_1)/(p_2) = eta T_0 (for eta = (p_1)/(p_2))`
or, `Delta T = eta T_0 - T_0 = (eta - 1) T_0` so, `Q_2 = (vR)/(gamma - 1).(eta - 1) T_0`
Thus, net heat transfer to the gas
`Q = vRT_0 1n eta + (vR)/(gamma-1).(eta - 1) T_0`
or, `(Q)/(vRT_0) = 1n eta + (eta - 1)/(gamma - 1)`, or, `(Q)/(vRT_0) - 1n eta = (eta - 1)/(gamma -1)`
or `gamma = 1 + (eta - 1)/((Q)/(vRT_0) - 1n eta) = 1+ (6.1)/(((80 xx 10^3)/(3 xx 8.314 xx 273)) - 1n 6) = 1.4`.