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3.012 Fundamentals of Materials Science
Fall 2003
Lecture 4: 09.15.03 Temperature, heat, and entropy
Today:
LAST TIME .............................................................................................................................................................................................. 2
DEFINING TEMPERATURE ......................................................................................................................................................................... 3
The zeroth law of thermodynamics .................................................................................................................................................... 3
CONSEQUENCES OF THE RELATION BETWEEN TEMPERATURE, HEAT, AND ENTROPY: HEAT CAPACITY .................................................... 4
The difference between heat and temperature ................................................................................................................................... 4
Defining heat capacity ....................................................................................................................................................................... 4
Examination of heat capacities in different materials........................................................................................................................ 5
The third law, absolute temperature, and heat capacities ................................................................................................................. 6
Calculations with heat capacities ...................................................................................................................................................... 8
REFERENCES ......................................................................................................................................................................................... 11
Reading:
Mortimer, Ch. 3 pp. 60-71; Ch. 4.5 pp. 119-122
H.A. Bent, The Second Law, Ch. 5: ‘Survey of Molar Entropies’, pp. 32-36
Supplementary Reading:
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Last time
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defining temperature

Can you define temperature?
The zeroth law of thermodynamics

In the first lecture, we introduced the concept that thermodynamics is ‘ruled’ by 4 empirical laws, and now we’ll
discuss another of these, called the zeroth law- which will help us to define temperature:
o Zeroth law:
B
A
C
o
The temperature is defined as that which is equal when heat ceases to flow between these
samples.
o
Last time, we saw that entropy changes are related to heat transfer in reversible processes. The first law
for a reversible process is:
dU  dqrev  dw  TdS  PdV
(Eqn 1)

For a process where no work is performed (isochoric process), dU = TdS. This gives us another
way to define temperature:

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Consequences of the relation between temperature, heat, and entropy: heat capacity
The difference between heat and temperature

Why do we need to define the property of temperature independently of heat? Why doesn’t the temperature of a
material simply quantify the amount of heat transferred to it?
o Looking again at the reversible process definition of entropy:
dqrev  TdS
(Eqn 3)
o
(Eqn 4)
T
or
dqrev
dS
(Eqn 3) tells us something about what entropy is: it is a function that relates heat transferred to the
temperature of a system. Let’s draw an analogy to energy transfer in the form of mechanical work:

dw  Fdx




…when a force F is applied to a system, the response of the system is the change in physical
dimensions dx.
In a similar manner, temperature may be thought of as a thermodynamic force, and entropy its
conjugate displacement (that aspect of the system that responds to the force) when no
mechanical work is being performed by the system.
Defining heat capacity

The following experiment can be done: Consider two roughly marble-sized balls of equal mass, one made of gold
and one made of aluminum, both heated to 100° C. The two balls are placed on a strip of wax suspended over
an open can. The Au ball melts through the strip of wax and falls into the can while the other ball does not.
o
Why would two metals heated to the same temperature have a different ability to melt the wax strip? The
answer lies in (Eqn 3), which says that two metals at the same temperature do not necessarily contain the
same amount of heat- since the amount of heat transferred into the balls is equal to the product of their
temperature and their entropy change dS. This is another outcome of the link between temperature, heat,
and entropy.
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

Fall 2003
Physcially, it is due to differences in composition, structure, and bonding between atoms and
molecules in different materials. We measure this difference in responses to heat transfer by
the thermodynamic property heat capacity.
Heat capacities are defined depending on the type of process a system is exposed to: constant-pressure or
constant volume.
o In a constant pressure process (e.g. heating a bar in a beaker on the lab bench), the heat capacity is:
dq 
S 
C p   rev   T 
 dT P
T P
(Eqn 5)
 …where we have used our reversible process definition for the entropy in the second equality.
For a constant volume process (e.g. heating a gas in a rigid chamber), the heat capacity is given by:
o

(Eqn 6)
o

dq 
S 
CV   rev   T 
 dT V
T V
Why should dq/dT at constant pressure differ from dq/dT at constant volume? This is a consequence of
the path dependence of q: the amount of heat transferred by processes following different ‘paths’
(process conditions) will be different.
Examination of heat capacities in different materials

Experimental heat capacities are often given in units of J/moleK (molar heat capacity) or J/gK (the latter are
called specific heat capacities or specific heats). Differences in heat capacity quantify differences in the entropy
change within materials in response to a change in temperature- and thus differences in the amount of heat
absorbed for a given temperature change. Heat capacities can vary widely among different materials: 2
Constant Pressure Heat Capacity at T = 298K
State of aggregation
CP (J/moleK)
C (diamond)
Solid
6.23
Fe
Solid
8.44
C (graphite)
Solid
8.573
Al
Solid
24.3

Au
Solid
25.4
Hg
Liquid
28.03
N2 (g)
Gas
29.124
O2
Gas
29.33
CO2 (g)
Gas
29.334
H2O (g) (unstable at 298K)
Gas
33.574
H2O (s - ice)
Solid
37.66
H2O (l)
Liquid
75.31
CH2OH
Liquid
112.9
Material
o
o
Cp (J/ gK)
0.519
0.444
0.713
0.899
0.129
0.139
1.04
0.916
0.667
1.865
2.092
4.184
3.64
Heat capacities can be very different between materials with similar bonding (e.g. two metals Au and Fe),
materials with the same atomic constituents but different atomic arrangements (e.g. diamond and
graphite), and between the same material in different states of aggregation (e.g. liquid water vs. ice).
Classical thermodynamics does not explain the differences in heat capacities experimentally observed.
 We will probe the molecular causes of these differences in the second half of the term when we
begin to explore bonding and statistical mechanics, which seeks to describe thermodynamic
properties that derive from molecular details of materials.
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SPEND MORE TIME LINKING TO STRUCTURE HERE
The third law, absolute temperature, and heat capacities
Temperature scales and the third law of thermodynamics

The Celsius scale with which we are most familiar in everyday life was derived by defining the temperature at
which water freezes as 0 degrees. Last time, we introduced the absolute (Kelvin) temperature scale- a scale with
a zero point and no possibility of a lower temperature- without much discussion of where it comes from. What
defines T = 0 K?
o The zero point of the absolute temperature scale was originally derived by identifying the temperature at
which heat would be converted into work with 100% efficiency.

The third law, like the other laws of thermodynamics, is derived from empirical observations made by scientists
studying the behavior of thermodynamic systems. The third law derived from experiments looking at the behavior
of heat capacities at lower and lower temperatures. One statement of the third law is:
o The third law:
o
There’s not much more to add to this right now- we will come back to the third law when we examine the
molecular origins of thermodynamic behavior in the second part of the term. We’ve stated (without trying
to demonstrate it) that entropy is a measure of disorder on the molecular scale- thus the third law says
that disorder is zero when a perfect crystal is taken to zero Kelvin. This agrees with the notion that heat
introduces entropy by vibrating, rotating, and translating molecules of a material.
Variation of heat capacity with temperature

Heat capacities also approach zero as the temperature approaches 0 Kelvin. However, they can often be
approximated as constant over experimentally relevant temperatures:
o Examples (from the texts by Zemansky and Gaskell):5
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Calculations with heat capacities

Knowledge of heat capacities allows us to perform calculations of entropy from simple experimental
measurements. Given either a value for the heat capacity (assumed constant over the temperature range of
interest) or an empirical formula for the dependence of the heat capacity with temperature, we can directly
compute absolute entropies.
o For example:


…Note that because the heat capacity approaches 0 at 0K, it would be unwise to assume CP is a
constant when integrating starting at T=0K.
Because the entropy is related to the integral of Cp, we find that the general trends in heat
capacity discussed above are also seen in the entropy/mole of materials:
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(Gaskell3)

o

Note the discontinuities in entropy at phase transitions (e.g. melting of solids to liquids) in
the diagram- we will examine these in more detail in the coming lectures.
For cases at finite temperatures where the heat capacity can be assumed to be relatively constant, we
can determine an entropy change for a change of temperature. For example, if we change the
temperature of an aluminum sample from T=293K to T=400K by a constant pressure process, we get:
We’ve so far only been able to calculate changes in internal energy for ideal gases using the first law combined
with the ideal gas law. The heat capacity gives us a means to determine changes in internal energy for arbitrary
materials. For a constant volume process:
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CV
T1
T2
U
T (K)
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References
1.
2.
3.
4.
5.
Carter, W. C. (2002).
Dill, K. & Bromberg, S. Molecular Driving Forces (New York, 2003).
Gaskell, D. R. Introduction to Metallurgical Thermodynamics (Hemisphere, New York, 1981).
Lupis, C. H. P. Chemical Thermodynamics of Materials (Prentice-Hall, New York, 1983).
Zemansky, M. W. & Dittman, R. H. Heat and Thermodynamics (McGraw-Hill, New York, 1997).
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