Two particular systems have the following equations of state: 1/T^((1)) =3/2 R N

Question

Two particular systems have the following equations of state: 1/T^((1)) =3/2 R N^((1))/U^((1)) and 1/T^((2)) =5/2 R N^((2))/U^((2)) where R is the gas constant. The mole number of the first system is N^((1))=2 and that of the second is N^((2))=3. The two systems are separated by a diathermal wall, and the total energy in the composite system is 2.5×?10?^3 J. What is the internal energy of each system in equilibrium?

I know that U^(1)+U^(2) at equilibrium will = 2.5x10^3J but I'm not sure how to work this through to the individual U's

Explanation / Answer

At equilibrium,
T1=T2=T
and,
(U1)'+(U2)'=2.5x10^3.... (i)
From ques,
(U1)'=3*(N1)*(.5*R*T)
(U2)'=5*(N2)*(.5*R*T)
N1=2, N2=3..... (no of moles remains constant in each system)
from (i),
.5*R*T = (2.5x10^3)/(3*2+5*3) =(2.5x10^3)/21

thus,
(U1)'=3*2*(2.5x10^3)/21=7.14 x 10^2 J
(U2)'=5*3*(2.5x10^3)/21=1.786 x 10^3 J

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