Page 97 - Essentials of physical chemistry

The First Law of Thermodynamics 59

thermodynamics is not complicated but more thinking is required as to the sign of the quantities.
Thermodynamics encourages thinking more with less automatic use of formulas.
We have mentioned the relationship U ¼ (3RT=2) and DU ¼ (3R=2)DT. For a Boltzmann
atomic gas, we can take the derivative with respect to temperature while holding the volume
constant, recognizing that the laboratory relationships would involve P, V, T, n so maybe at least
one other variable is involved assuming that knowledge of two of the three variables would
determine the third under some equation of state for a fixed amount (moles) of gas:

qU qU qU
dV,
dU ¼ dT þ dV ¼ C v dT þ
qT qV qV
V T T
assuming constant moles.
qU
. This also brings up the realization that for
Thus, we introduce the new quantity C V
qT
qU qU V
an ideal gas we have ¼ ¼ 0, since the energy of the ideal gas only depends on
qV qP
T T
temperature. In fact we can enlarge the definition of an ideal (monoatomic) gas to mean:
Ideal Gas
1. PV ¼ nRT

qU qU
2. ¼ ¼ 0
qV qP
T T
We note that these conditions of energy independence from T and P may not be true for real gases or
even for the van der Waals gas but they are a reasonable approximation at low pressure and high
temperature where the ideal gas equation is good.
ENTHALPY AND HEAT CAPACITIES

Next we come to a practical problem: What happens even in household kitchens when a sealed container
is opened? Soft drinks are under some pressure in their containers, so pressure is released when the
container is opened. Old fashioned glass milk bottles brought in from the cold would often pop out their
cardboard top plugs when warmed in a kitchen environment due to the Charles–Gay-Lussac relation-
ship. Thus, we see opening storage containers involves a small but nonzero pressure push from the
interior of a container against the nominal one atmosphere pressure in the environment. This relatively
small pressure change occurs in the laboratory as well for every process that is open to the atmosphere
and we know that any PV product is energy, even if it is small. Thus, it has proven practical to define
another energy quantity called the ‘‘enthalpy,’’ which is defined so that the atmospheric pressure and any
volume change due to ambient conditions is added to the internal energy as a definition:

H U þ PV:
Note that H is also a state variable because the PV product is energy and U is already a state
variable. This does make some things more complicated even though the new terms may be small in
magnitude so that we now have:
dH ¼ dU þ PdV þ VdP:

Again we appeal to practical conditions in that the external atmospheric pressure is essentially
constant depending only on the weather variations. The atmosphere of the planet is so huge that
opening a pressurize vessel or a vacuum chamber will not change the pressure of the atmosphere.
Thus, we see that to a good approximation:
dH ¼ dU þ PdV þ 0: