Subsystem 1 has a mass M1, and is initially a solid having a specific heat cs. I

Question

Subsystem 1 has a mass M1, and is initially a solid having a specific heat cs. It is initially at temperature T1i Tifp, where Titp is the freezing point temperature of Subsystem 1. When it is placed in thermal contact with Subsystem 2 having a mass M2, specific heat c2 and initial temperature T2i, Subsystem 1 melts completely into its liquid phase having a specific heat of cu. The system eventually reaches an equilibrium temperature of Tea: The specific latent heat of fusion of Subsystem 1 is Ln.Assume that the system comprised of the two subsystems is thermally isolated from the rest of the universe. Derive an algebraic expression for the equilibrium temperature, Tea, of the system.

Explanation / Answer

mass of subsystem 1 = M1
initially solid
specific heat = c1s
Initial temp = T1i < T1fp
T1fp = freezing point of subsystem 1
liquid phase specific heat = c1l
specific latent heat of fusion = Lf1

subsystem 2
Mass = M2
specific heat c2
initial temp = T2i

equilibrium temp = Teq
so heat transferred to system 1 = heat transferred from system 2
M1*c1s(T1fp - T1i) + M1*Lf1 + M1c1l(Teq - T1fp) = M2*c2*(T21 - Teq)
M1c1l*Teq + M2*c2*Teq = M2*c2*T21 + M1c1l*T1fp - M1*c1s(T1fp - T1i) - M1*Lf1
Teq = [M2*c2*T21 + M1c1l*T1fp - M1*c1s(T1fp - T1i) - M1*Lf1]/(M1cl1 + M2c2)

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