36 PARTI THERMODYNAMICS AND KINETICS
(db2.6)
The change in internal energy can then be written in terms of the internal pressure Pinternal
and specific heat at constant volume:
dU=PinternaldV+CVdT (db2.7)
For an ideal gas, since Pinternalis zero, the specific heat at constant volume represents the change
in the internal energy in response to a change in temperature. Although CVis determined
at constant volume, it can be considered to be a coefficient that determines how much effect
a change in temperature will have on the internal energy. When nonideal conditions are
included, the change dUis modified according to Pinternal. For real gases, if attractions among
molecules are found to dominate, then Pinternalwill be positive and if repulsion interactions
dominate then this term will be negative.
Similar expressions can be written for the other thermodynamic parameters. For example,
consider the change in enthalpy at constant volume. In this case the remaining parameters
are temperature and pressure, and the change in enthalpy can be written in terms of their
partial derivatives:
(db2.8)
Of these two partial derivatives, the first is the change in enthalpy due to a temperature change
at constant pressure, which was shown to be equal to the specific heat at constant pressure
(eqn db2.3). The second term, the change in enthalpy in response to a pressure change, can
be written in terms of the product of two derivatives involving the temperature following
the guidelines for the chain relationship of partial derivatives. Of these two derivatives, the
second is just the specific heat at constant pressure and the first is called the Joule–Thompson
coefficientand denoted by μ:
(db2.9)
The Joule–Thompson coefficient is commonly used in understanding how refrigerators work
and the process of liquefaction of gases. Refrigerators work with non-ideal gases and take
advantage of attractive forces between molecules (and the negative value of μ). The gases
used in refrigerators release heat when gas molecules come together, and take up heat when
gas molecules are pulled apart. Thus, when a gas is allowed to expand at constant pressure,
the temperature of the gas will drop according to the coefficient. In a refrigerator, a gas is
cyclically allowed to expand and contract to move heat from inside to outside a system. The
Joule–Thompson effect is also used to liquefy gases, as a gas is initially poised at high pressure
and is then allowed to expand and hence cool. By performing the volume change cyclically
the temperature will systematically drop until the gas condenses to a liquid.
∂
∂
∂
∂
∂
∂
H
P
T
P
H
THPT
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ =−
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ =−μCP
ddH
HPT
T
T
HPT
P P
(,)(,)
=
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ +
⎛
⎝
⎜⎜
⎞
⎠
∂
∂
∂
∂
⎟⎟⎟
T
dP
∂
∂
U
V
a
V
n
P
T
internal
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ =
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ =
2
