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Three moles of an ideal gas at 300 K are isothermally expanded to five times its volume and heated at this constant volume so that the pressure is raised to its initial value before expansion. In the whole process 83.14 KJ heat is required. Calculate the ratio C_p//C_v of the gas. log. 5 = 1.61 and R = 8.3 Joules/mole K.

Answer» <html><body><p></p>Solution :For the isothermal expansion, by Boyle.s law, we have<br/> `P_1 V_1=P_2 V_2or(P_1)/(P_2) = (V_2)/(v_1) = (5V_1)/( V_1) = 5`<br/>Now, applying gas law at constant volume, we <a href="https://interviewquestions.tuteehub.com/tag/get-11812" style="font-weight:bold;" target="_blank" title="Click to know more about GET">GET</a> <br/> `(P_2)/(T_2)=(P_3)/(T_3)( thereforeP_3 = P_1)(or)(P_2)/(T_2) =(T_2)/(T_3) = (T_1)/(T_3)` <br/> (As <a href="https://interviewquestions.tuteehub.com/tag/first-461760" style="font-weight:bold;" target="_blank" title="Click to know more about FIRST">FIRST</a> expansion was isothermal, <a href="https://interviewquestions.tuteehub.com/tag/hence-484344" style="font-weight:bold;" target="_blank" title="Click to know more about HENCE">HENCE</a> `T_1 =T_2`.) <br/> `threfore(T_3)/(T_1) =(P_1)/(P_2) = 5or T_3 = 5T_1 = 5 xx300= 1500K` <br/>Now ` dQ= dU+ <a href="https://interviewquestions.tuteehub.com/tag/dw-943287" style="font-weight:bold;" target="_blank" title="Click to know more about DW">DW</a>= nC_vd Tint_( V_i)^(v_2) PdV ` <br/>But `P= nRT_1 //V`, where `mu`is the number of moles of the gas.<br/> ` thereforedQ= nC _v dT + nRT_1log _e(V_2 //V_1)` <br/>Now `C_p-C_v=Rand(C_p -C_v) = gamma. so C_v = R // ( gamma -1)` <br/> `therefored Q=( nR)/( gamma -1) dT+ nRT_1log_e(V_2)/(v_1)`<br/><a href="https://interviewquestions.tuteehub.com/tag/substituting-1231652" style="font-weight:bold;" target="_blank" title="Click to know more about SUBSTITUTING">SUBSTITUTING</a> the values : `83.14 xx 10^3` Joule `=( 3 xx 8.3 )/( gamma -1)` <br/>`(1500- 300 ) +8.3300log_e5 (or)83140 =( 29880)/( gamma -1) + 12027( log_e 5=1.61 ]thereforegamma= 1.42`</body></html>


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