Lecture 5

Question 1

Refrigerators

Answer 1

Running a heat engine in reverse creates a refrigerator.

Question 2

the efficiency of a refrigerator is typically defined as

Answer 2

η(R) = Q2/W = Q2/Q1-Q2

Question 3

for a Carnot refrigerator the efficiency is defined as

Answer 3

ηC(R) = T2/T1-T2

Question 4

heat pumps efficiency is defined as

Answer 4

ηc(HP) = Q1/Q1-Q2 = T1/T1-T2

Question 5

Clausius inequality

Answer 5

Consider:

Principle reservoir at T0

A system undergoing a cyclic process by N incremental steps of variable temperature

some external ‘principal’ thermal reservoir

system interacts with the environment via N Carnot engines, mediating all incremental heat transfer to and from the system, and each doing or taking work as required.

Applying the first law to individual engines and the composite system of all the engines and system together.

Question 6

the Clausuis inequality for a cyclic process (formula)

Answer 6

∮ δQ/T ≤ 0

Question 7

the Clausuis inequality for a reversible cyclic process (formula)

Answer 7

∮R δQ/T = 0

Question 8

Processes that are not isolated can

Answer 8

lose heat to their environments

Question 9

A derivation of the Clausius inequality for the surroundings, that is the principle reservoir can be sketched in the following way:

Answer 9

As there is only a single reservoir, Q < 0 according to the Kelvin Planck formulation of the second law. This means cyclic processes tend to dump heat.

Q = Σ δQi < 0 and T0 > 0 -> ∮ δQ/To ≤ 0

the entropy change of the surroundings depends on the heat transferred and the surrounding temperature. For a closed system we always need to consider the entropy of the surroundings which can be expanded to the whole universe as the ultimate closed system.

which leads to the ‘real’ Clausius inequality

∮ δQ/To ≤ 0