Question

Two thermally insulated vessels 1 and 2 are filled with air at temperature (T1, T2), volume (V1, V2) and pressure (P1, P2) respectively. If the value joining the two vessels is opened, the temperature inside the vessel at equilibrium will be

A

T1 +T2

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B

(T1 + T2)/2

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C

T1T2(P1V1+P2V2)/P1V1T2+P2V2T1)

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D

T1T2(P1V1+P2V2)/P1VT1+P2V2T2)

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Solution

The correct option is C T1T2(P1V1+P2V2)/P1V1T2+P2V2T1) According to the kinetic theory, the average kinetic energy(KE) per molecule of a gas = 32KT. Let n1 and n2 be the number of moles of air in vessels 1 and 2 respectively. Before mixing, the total KE of molecules in the two vessels is E1 = 32n1kT1 + 32n2kT2 = 32k(n1T1 + n2T2) After mixing, the total KE of molecules is E2 = 32(n1 + n2)kT Where T is the temperature when equilibrium is established. Since there is no loss of energy (Because the vessels are insulated), E2 = E1 or 32(n1 + n2)kT = 32k(n1T1 + n2T2) or T = n1T1 + n2T2n1 + n2 Now P1V1 = n1RT1 and P2V2 = n2RT2 which give n1 = P1V1RT1 and n2 = P2V2RT2 Using these in Eq. (1) and simplifying, we get T = T1T2(P1V1 + P2V2)(P1V1T2 + P2V2T1)

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