a) What is the temperature of the solid? b) What is the entropy of the solid? c)

Question

a) What is the temperature of the solid?

b) What is the entropy of the solid?

c) An identical "solid" with the same spectrum and number of oscillators, but with energy E = 0, is brought into a weak contact with the first without adding any extra energy to the combined system. Determine the entropy change long after the contact is established, and the new temperature.

d) Hard, optional. Imagine that you performed step (c) reversibly, so that a new entropy change was actually zero. In principle, it is possible to do this. For example, by running a "heat engine" between the two solids with an appropriately chosen cycle. Analye this scenario to determine the energy E and the temperature T in the final state. How much mechanical work have you removed (or added) from (to) the system?

Explanation / Answer

Answer:

N = 1023 Oscillators

Given information is

E = qN and N = q

(a) Temperature of solid = (del S/ del E)-1 = (Nk/E)-1 = (Nk/3qN)-1 = (Nk/3N2)-1

= 3N/k = 3 x1023/(1.38 x 10-23) = 2.17 K

(b) Entropy S = k ln(omega)

S = k [ ln(q+N)! - lnq! -lnN! ]

S = k [ ln(2N)! - lnN! -lnN! ] given that N = q

= k [ ln(2N)! - 2lnN! ] = 1.38 x 10-23 [ ln(2x1023)! - 2 ln (1023)!]

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