Two identical ideal gases with the same pressure P and the same number of partic

Question

Two identical ideal gases with the same pressure P and the same number of particles N = 1023, bit with different temperatures T1 = 320K and T2 = 340K, are confined in two vessels of volum,e V1 and V2 which are then connected. Find the change in entropy after the system has reached equilibrium

Explanation / Answer

Finding final temperature, Tf Energy balance, Let assume T2 > T1 and V2 > V1 Q1 = Q2 m1*cp*(Tf - T1) = m2*cp* (T2-Tf) Tf - T1 = T2-Tf Tf = T1+T2 / 2=320+340/2=330k Entropy change for first vessel, Tds = du + pdv ds = du/T + (p/T)dv, du = cvdT, p/T = R/v ?S1 = n [cv*ln(Tf/T1) + Rln(Vf/V1) ?S1 = n [cv*ln(T1+T2/2T1) + Rln(V1+V2 / V1) Entropy change for second vessel, ?S2 = n [cv*ln(Tf/T2) + Rln(Vf/V2) ?S2 = n [cv*ln(T1+T2/2T2) + Rln(V1+V2 / V2) Total entropy change, ?S = ?S1 + ?S2 ?S = n*{[cv*ln(T1+T2/2T1) + Rln(V1+V2 / V1) + [cv*ln(T1+T2/2T2) + Rln(V1+V2/V2) } ?S = n*{cv* [ ln(T1+T2/2T1) + ln(T1+T2/2T2) ] + R [ ln(V1+V2 / V1) + ln(V1+V2/V2) ] } ?S = n * { cv * [ ln(T1+T2)^2 / 4T1T2) + R [ ln(V1+V2)^2 / V1V2) ] } Because, N = n*Na, Na = Avogadro's constant = 6.02 * 10^23 mol^-1, then ?S = (N/Na) * { cv * [ ln(T1+T2)^2 / 4T1T2) ] + R [ ln(V1+V2)^2 / V1V2) ] } mere substitution.

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