Physical Chemistry Third Edition

3.1 The Second Law of Thermodynamics and the Carnot Heat Engine 111

announced after the other laws. The order of announcement was as follows: the second

law, followed by the first law, the third law, and the zeroth law. The name “zeroth law”

Ralph H. Fowler, 1889–1945, was a was coined by Ralph H. Fowler.
prominent professor of physics at
Cambridge University.

Any other reservoir at the same temperature could be substituted for one of the

reservoirs of our Carnot engine without any change in our analysis. The value of

the ratioq 3 /q 1 in a Carnot cycle therefore depends only on the temperatures of the

reservoirs and not on their composition. The thermodynamic temperature, θ,is

defined by

θc

θh



∣

∣

∣

∣

q 3

q 1

∣

∣

∣

∣ (definition ofθ) (3.1-13)

whereθcis the thermodynamic temperature of the cold reservoir andθhis the thermo-

dynamic temperature of the hot reservoir. We require the thermodynamic temperature

to be positive. The Carnot efficiency is now given by

ηCarnot 1 −

θc

θh

(3.1-14)

The thermodynamic temperature scale is not related to any particular kind of substance

and is therefore more fundamental than the ideal gas temperature scale. We now show

that the thermodynamic temperature scale can coincide with the ideal gas temperature

scale. Assume that the working fluid of a Carnot engine is an ideal gas with a constant

heat capacity. For the first step of the Carnot cycle, from Eq. (2.4-10)

q 1 nRThln(V 2 /V 1 ) (3.1-15)

For the third step,

q 3 nRTcln(V 4 /V 3 ) (3.1-16)

We can now locate the states at which steps 1 and 3 terminate by using Eq. (2.4-21)

twice:

Tc

Th



(

V 2

V 3

)nR/CV, m

(3.1-17)

and

Tc

Th



(

V 1

V 4

)nR/CV, m

(3.1-18)

These two equations imply that

V 1

V 2



V 4

V 3

(3.1-19)

When this relation is substituted into Eq. (3.1-16),

q 3 nRTcln(V 4 /V 3 )nRTcln(V 1 /V 2 )−nRTcln(V 2 /V 1 ) (3.1-20)

Equation (3.1-20) and Eq. (3.1-15) are substituted into Eq. (3.1-5) to give

wcycle−qcycle−q 1 −q 3 nR(−Th+Tc)ln(V 2 /V 1 ) (3.1-21)