1549380323-Statistical Mechanics Theory and Molecular Simulation

Laws of thermodynamics 61

Eqn. (2.2.18) indicates that there is a quantity ∆Qrev/T whose change over the
closed cycle is 0. The “rev” subscript serves as a reminder that theCarnot cycle is
carried out using reversible transformations. Thus, the quantity∆Qrev/Tis a state
function, and although we derived this fact using an idealized Carnotcycle, it turns
out that this quantity is always a state function. This means that there is an exact
differential dS= dQrev/Tsuch thatSis a state function. The quantity ∆S, defined
by

∆S=

∫ 2

1

dQrev

T

, (2.2.19)

is therefore independent of the path over which the transformation from state “1” to
state “2” is carried out. The quantitySis the entropy of the system.
The second law of thermodynamics is a statement about the behavior of the en-
tropy in any thermodynamic transformation. From eqn. (2.2.10), which implies that
dQirrev<dQrev, we obtain

dS=

dQrev

T

>

dQirrev

T

, (2.2.20)

which is known as theClausius inequality. If this inequality is now applied to the
thermodynamic universe, an isolated system that absorbs and releases no heat (dQ=
0), then thetotalentropy dStot= dSsys+ dSsurrsatisfies

dStot≥ 0. (2.2.21)

That is,in any thermodynamic transformation, the total entropy of the universe must
either increase or remain the same.dStot>0 pertains to an irreversible process while
dStot= 0 pertains to a reversible process. Eqn. (2.2.21) is the second law of thermo-
dynamics.
Our analysis of the Carnot cycle allows us to understand two equivalent statements
of the second law. The first, attributed to William Thomson (1824–1907), known later
as the First Baron Kelvin or Lord Kelvin, reads:There exists no thermodynamic trans-
formation whose sole effect is to extract a quantity of heat from a high-temperature
source and convert it entirely into work. In fact, some of the heat absorbed atThis
always lost in the form ofwaste heat, which in the Carnot cycle is the heatQlreleased
atTl. The loss of waste heat means that−Wnet<−Whor that the net work done
by the system must be less than the work done during the first isothermal expansion
phase.
Now suppose we run the Carnot cycle in reverse so that an amount of heatQlis
absorbed atTland released asQhatTh. In the process, an amount of workWnetis
consumedby the system. Thus, the Carnot cycle operated in reverse performs as a
refrigerator, moving heat from a cold source to a hot source. Thisbrings us to the
second statement of the second law, attributed to Clausius:There exists no thermo-
dynamic transformation whose sole effect is to extract a quantity of heat from a cold
source and deliver it to a hot source. That is, heat does not flow spontaneously from
cold to hot; moving heat in this direction requires that work be done.