Page 127 - Essentials of physical chemistry

The Second and Third Laws of Thermodynamics 89

is constant (at atmospheric pressure) in a laboratory setting unless the system is sealed. Thus,
J.W. Gibbs (1839–1903) defined what is now known as the ‘‘Gibbs free energy, G,’’ which has
proved to be very useful. Gibbs defined G H TS and once again all the terms are in terms of
energy units. This time we add and subtract VdP as well as SdT to the first law:

dU ¼ TdS PdV þ SdT Sdt þ VdP VdP ¼ d(TS) SdT d(PV) þ VdP
so that we find

dU þ d(PV) d(TS) ¼ SdT þ VdP ¼ d(U þ PV TS) ¼ d(H TS) ¼ dG

and so we obtain

dG ¼ SdT þ VdP:

The equation of G ¼ H TS is a better indicator of an equilibrium so that when dT and, especially,
when dP are zero, then dG ¼ 0 indicating an equal trade-off between increasing entropy and
decreasing energy. Next, consider the definition of H U þ PV and use the two conditions
above to find that that there is a fourth equation for dH as found from

dH ¼ dU þ PdV þ VdP ¼ (TdS PdV) þ PdV þ VdP ¼ TdS þ VdP:

We have just rushed over more than 100 years of developments in thermodynamics by focusing on
key equations, and we can consolidate the ‘‘essential’’ knowledge for this text as

qT qV
dH ¼ TdS þ VdP and ¼ ,
qP qS
S P

qT qP
dU ¼ TdS PdV and ¼ ,
qV qS
S V

qS qV
dG ¼ SdT þ VdP and ¼ ,
qP qT
T P

qS qP
dA ¼ SdT PdV and ¼ :
qV qT
T V
This is pretty much the jackpot of equations for thermodynamics and now reveals why we chose to
prove the Carnot cycle relationship so that we could get to dq rev ¼ TdS. Note the alternating pattern
of signs and the variables when assembled in the order ‘‘HUGA,’’ which may help organize the
equations in your mind. We could spend a lot more time on the history of the Helmholtz and Gibbs
free energy derivations, but in this course on ‘‘essentials,’’ we swam through a narrow intellectual
cave (Carnot cycle) to reach a beautiful expansive blue grotto of valuable knowledge with these
eight equations. The four auxiliary partial derivative equations are called the ‘‘Maxwell relation-
ships.’’ They result from the fact that H, U, G, and A are all state variables with ‘‘exact differentials.’’
Since it does not matter, which of the two variables change in either order for the basic four
‘‘HUGA’’ equations, we can do the second derivatives in either order.
2
qH q H qT
dH ¼ TdS þ VdP so ¼ T and then ¼ where we have used the
qS qP qS qP
P S
convention that the most recent derivative is with respect to the lower left variable in the second
derivative.