Lecture 4

Question 1

A heat engine works by

Answer 1

a heat injection from the warm reservoir and a heat rejection to the cold reservoir.

Question 2

Proof of Carnot’s theorem

Answer 2

Consider two engines working between heat reservoirs at temperatures T1 and T2.

One engine produces work W which is used to run the other in reverse.

Q2(A) = Q1(A) - W
Q2(B) = Q1(B) - W

take the reversed engine to be an optimal Carnot engine, that is we can reverse the arrows without changing the ratio W/Q1, but suppose the other engine is more efficient then

η(A) > η(B) -> W/Q1(A) > W/Q1(B) -> Q1(B) > Q1(A)

if this is true then,

Qfrom cold = Q2(B) - Q2(A) = Q1(B) - Q1(A) > 0

that is transfer of heat from a cold body to a hot body without work, which violates the Clausius form of the second law. Therefore engine A cannot be more efficient than the Carnot engine

Question 3

Corollary

Answer 3

all Carnot engines operating between the same two temperatures reservoirs have the same efficiency

Question 4

η is independent of the working substance and internal workings of the engine and solely dependent on the reservoir temperatures (formula)

Answer 4

η = W/Q1 = Q1-Q2/Q1 = 1 - Q2/Q1

with

Q2/Q1 = f(T1,T2)

Question 5

the thermodynamic temperature is (formula)

Answer 5

T(K) = 273.16K Q/Q(TP water)

Question 6

the efficiency of a Carnot engine can be reformulated in terms of the temperature ratio

Answer 6

ηc = 1- T2/T1