Question

Two solid bodies at initial temperatures T1 and T2, with T1 > T2, are placed in thermal contact with each other. The bodies exchange heat only with eachother but not with the environment. The heat capacities C ≡ Q/∆T of each body are denoted C1 and C2, and are assumed to be positive.

(a) Is there any work done on the system? What is the total heat absorbed by the system? Does the internal energy of each subsystem U1 and U2 change? What about the total internal energy U = U1 + U2?

(b) Calculate the final temperature, Tf , of the system.

(c) Show that T1 > Tf > T2.

(d) With the result of part (b), discuss what happens if C1 = 0 or C2 = 0. What about if C1 → ∞ or C2 → ∞? Interpret your results

Answer 1

(a)There is no work done on the system because here system is neither expanded nor contracted also not displaced so net amount of work done equal to zero i.e.

    

according to the principle of thermodynamics -

Heat gained by(absorbed by) cold body=-Heat lost by hot body

Since heat lost and heat gain both are equal so for the whole system total heat absorbed by system will be also zero.

Since temperature of hot body decrease and cold body increases and also there is no work done so we can say that internal energy of hot body decrease while for cold body it increases.

According to first law of thermodynamics-

since work done is zero here so-

for hot body

   -------------------------------(1)

(negative sign is used because it losses heat)

Similarly for cold body

---------------------------------(2)

adding equ(1) and equ(2)-

so there is no change in the internal energies of system so we can say that sum of internal energy remains constant during the process i.e.

---------------------------------------------------------------------------------------------------------------------------------

(b) Heat lost by hot body      --------------------------(3)

heat gained by cold body    --------------------------(4)

according to principle of caloriemetry both will be equal-

--------------------------------------------------------------------------------------------------

(C)from equation(3) we can write-

since Q and C₁ Both are positive here so it is clear that-

------------------------------(5)

similarly from eqution (4)we can write-

since Q and C₂ Both are positive so-

     -------------------------------(6)

combine the relations of eequation(5) and (6)

--------------------------------------------------------------------------------------------

(d)if C₁ and C₂ both are zero it means they afre perfectly insulator and does not exchange when brought into contact so there temperature remains same i.e. T1 for first body and T2 for second body and there is no common final temperature exists.

If we put C1=C2=0 in the relation obtained in part (b) then we see that

which is indeterminate form so no final temperature exists.

if C1 and C2 both are infinite then they are ideal conductor.

by the formula obtained in part(b)

dividing by C1C2 in both nemerator and denomenator

since C1 and C2 both are infinite so above equation becomes.

which is also indeterminate form.

since infinite heat capacity means hot body can give infinite heat energy to cold body without drop in its teperature and reversely cold body can take infinite amount of heat energy without any rise in its temperature.so again their temperature remains same so no final temperature exists.