Some General Remarks Concerning Heat Capacity

CHEM 331
Physical Chemistry
Fall 2014
Some General Remarks Concerning Heat Capacity
The Heat Capacity is one of those quantities whose role in thermodynamics cannot be
underestimated. Measured using simple calorimetric measurements, it finds itself useful in
calculations involving any number of thermodynamic "potentials". But, more on this later. Here
we derive a few simple results involving the heat capacity which we will find useful as we
proceed with our discussion of thermodynamics.
The beautiful work of Joseph Black on calorimetry, the measurement of heat changes, was
published in 1803, four years after his death. In his Lectures on the Elements of Chemistry, he
pointed out the distinction between the intensive factor, temperature, and the extensive factor,
quantity of heat. Black showed that equilibrium required an equality of temperature and did not
imply that there was an equal "quantity of heat" in different bodies.
He then proceeded to investigate the capacity for heat or the amount of heat needed to increase the
temperature of different bodies by a given number of degrees.
It was formerly a supposition that the quantities of heat required to increase the heat of
different bodies by the same number of degrees were directly in proportion to the
quantity of matter in each … But very soon after I began to think on this subject (Anno
1760) I perceived that this opinion is a mistake, and that the quantities of heat which
different kinds of matter must receive to reduce them to an equilibrium with one another,
or to raise their temperatures by an equal number of degrees, are not in proportion to the
quantity of matter of each, but in proportions widely different from this, and for which no
general principle or reason can yet be assigned.
In explaining his experiments, Black assumed that heat behaved as a substance, which could flow
from one body to another but whose total amount must always remain constant. This idea of heat
as a substance was generally accepted at that time. Lavoisier even listed caloric in his "Table of
the Chemical Elements." In the particular kind of experiment often done in calorimetry, heat
does, in fact, behave much like a weightless fluid, but this behavior is the consequence of certain
special conditions. Consider a typical experiment: A piece of metal of mass m2 and temperature
T2 is introduced into an insulated vessel containing a mass m1of water at temperature T1. We
impose the following conditions: (1) the system is isolated from its surroundings; (2) any change
in the container itself can be neglected; (3) there is no change such as vaporization, melting, or
solution in substance, and no ne temperature T somewhere between T1 and T2, and the
temperatures are related by an equation of the form
c2 m2 (T2 - T) = c1m1 (T -T1)
Here, c2 is the specific heat of the metal and c2 m2 = C2 is the heat capacity of the mass of metal
used. The corresponding quantities for water are c1, and c1 m1 = C1. The specific heat is the heat
capacity per unit mass.
More careful measurements showed that the specific heat was itself a function of temperature. …
[Thus,] the heat capacity, being a function of temperature, should be defined precisely on in terms
of differential heat flow Q and temperature change. Thus, in the limit, [it] becomes
Q = C dT or C =
Physical Chemistry, 4th Ed.
Walter J. Moore
So, in general, measured under a given constraint x, the Heat Capacity of a substance is defined
as:
Cx
=
We have already specified two such constraints, constant volume and constant pressure, giving
us the heat capacities Cv and Cp, respectively. These two heat capacities are the most useful
experimentally and are related to the the internal energy U and to the enthalpy H; again,
respectively.
Cv
=
=
Cp
=
=
Since Cv and Cp can both be measured for a given substance, it is not unreasonable that they
should be related. It is this relationship which we now seek to establish.
We start with our expressions for dU:
dU
= Q + W
= Q - Pop dV
= Q - P dV
and
dU
=
dT +
= Cv dT +
dV
dV
Equating these expressions, we have:
Q - P dV = Cv dT +
dV
Dividing by dT and restricting ourselves to constant P:
- P
= Cv
+
Recognizing the lead term is Cp gives us:
Cp - P
= Cv +
Rearranging a bit gives us the desired result:
Cp - Cv =
In general we find that Cp > Cv. This is not unreasonable. When heat enters a system at constant
volume, it increases the "chaotic" motion of the system's constituents. This leads to an increase
in the temperature. (What we mean by "chaotic" motion is the translational, rotational and
vibration motion of the molecules. The first of these contributes to the kinetic energy of the
molecules and thereby increases the system's temperature.) When heat enters a system at
constant pressure, the system's volume will increase, doing work in the surroundings. The
molecules of the system will be "pulled" apart requiring energy. And, there will be an increase
in the "chaotic" motion of the system's constituents. Only this last will lead to an increase in
temperature, for which there is less heat energy available. For a given amount of heat, there are
more modes to absorb the energy without a temperature increase, if the heating is done at
constant pressure rather than at constant volume. Hence, Cp - Cv > 0.
Work to Surroundings
Heat
Inc Chaotic Motion
(Temperature)
Heat
"Pull" Molecules Apart
Inc Chaotic Motion
(Temperature)
In our above expression for Cp - Cv, the term:
P
can be thought of as the work against the external pressure. The term:
is the work done against the internal pressure. Recall, the internal pressure of a system is:
T =
For an Ideal Gas, we have:
= 0
and
V =
So,
=
This gives us the important result:
Cp - Cv =
= [P + 0]
(You can only use Cp - Cv = R for an Ideal Gas.)
Typically, for liquids and solids we find:
Cp ≅ Cv
This is illustrated by the heat capacity data for Water.
Cp (J/K mol) Cv (J/K mol)
75.3
74.8
= R