9.13 Free Energies
359
dH = T dS + V dp +
k
μ k dN k .
(9.89)
This relation tells us that if the enthalpy is expressed in terms of entropy, pressure,
and the amount of each substance in the system, then the thermodynamics of the
system is completely determined. For this reason, the set {S, p, N} are the ‘natural
variables’ for the enthalpy function.
The enthalpy change is used in chemistry to describe the heat energy released (or
absorbed) during a reaction for the following reason. Often, chemical reactions in a
laboratory test tube and in the cells of our body occur at constant pressure (usually
atmospheric). Under constant pressure conditions, the change in the enthalpy
(H ≡ H f − H i ) after a reaction that releases an amount of heat Q will be
H = E + ppV
= −Q − ppV + ppV
= −Q .
(9.90)
The minus sign on Q comes from the convention that heat into a system is positive.
Thus, under isobaric conditions, the negative of the heat released will be the change
in the enthalpy for the system.
As a system tends toward thermal equilibrium at constant pressure, the enthalpy
will decrease toward a minimum. This can be seen from the first law in the form
δE = δQ − pδV +
μ k δN k so, at constant pressure,
δH = δQ +
k
μ k δN k ≤ T dS +
k
μ k dN k
(9.91)
which gives δH ≤ dH .
Under constant pressure and fixed particle numbers, the enthalpy change H of
a system as the temperature changes can be found by measuring the system’s molar
specific heat (at constant pressure) c p :
H | p =
dH | p =
T
∂S
∂T
p
dT = n
c p dT ,
(9.92)
where n is the number of moles of substance in the system. Starting at a standard
state (denoted by a superscript (− −)), the enthalpy at temperature T will be
H (T ) = H
− −
+ n
T
T o
c p dT .
(9.93)
359
dH = T dS + V dp +
k
μ k dN k .
(9.89)
This relation tells us that if the enthalpy is expressed in terms of entropy, pressure,
and the amount of each substance in the system, then the thermodynamics of the
system is completely determined. For this reason, the set {S, p, N} are the ‘natural
variables’ for the enthalpy function.
The enthalpy change is used in chemistry to describe the heat energy released (or
absorbed) during a reaction for the following reason. Often, chemical reactions in a
laboratory test tube and in the cells of our body occur at constant pressure (usually
atmospheric). Under constant pressure conditions, the change in the enthalpy
(H ≡ H f − H i ) after a reaction that releases an amount of heat Q will be
H = E + ppV
= −Q − ppV + ppV
= −Q .
(9.90)
The minus sign on Q comes from the convention that heat into a system is positive.
Thus, under isobaric conditions, the negative of the heat released will be the change
in the enthalpy for the system.
As a system tends toward thermal equilibrium at constant pressure, the enthalpy
will decrease toward a minimum. This can be seen from the first law in the form
δE = δQ − pδV +
μ k δN k so, at constant pressure,
δH = δQ +
k
μ k δN k ≤ T dS +
k
μ k dN k
(9.91)
which gives δH ≤ dH .
Under constant pressure and fixed particle numbers, the enthalpy change H of
a system as the temperature changes can be found by measuring the system’s molar
specific heat (at constant pressure) c p :
H | p =
dH | p =
T
∂S
∂T
p
dT = n
c p dT ,
(9.92)
where n is the number of moles of substance in the system. Starting at a standard
state (denoted by a superscript (− −)), the enthalpy at temperature T will be
H (T ) = H
− −
+ n
T
T o
c p dT .
(9.93)