3.012 Fundamentals of Materials Science
Fall 2003
Lecture 6: 09.19.03 Measuring thermodynamic quantities in the laboratory
Today:
LAST TIME ............................................................................................................................................................ 2
DIRECT AND INDIRECT MEASUREMENT ................................................................................................................. 3
Calorimetry: Heat capacities and the measurement of internal energy and enthalpy ....................................................................... 3
Measuring expansion and compression ............................................................................................................................................. 6
MAXWELL RELATIONS ......................................................................................................................................... 9
REFERENCES ....................................................................................................................................................... 10
Reading:
Supplementary Reading:
-
Lecture 6 – measuring thermodynamic quantities
1 of 10
2/17/16
3.012 Fundamentals of Materials Science
Fall 2003
Last time

What
Lecture 6 – measuring thermodynamic quantities
2 of 10
2/17/16
3.012 Fundamentals of Materials Science
Fall 2003
Direct and indirect measurement

We are building toward the ability to predict how materials will change in response to thermodynamic forces (heat
and different forms of work). The functions internal energy and entropy will be used to calculate the equilibrium
state – the state the system tends toward in the presence of a given set of thermodynamic forces. However, a big
question you may have already wondered about is: How do we measure internal energy? Is there an ‘entropy
meter’?
o Some thermodynamic quantities are directly measurable, such as:
 Temperature (e.g. thermometer)
 Pressure (e.g. pressure gauge)
 Volume (e.g. graduated cylinder)
 Number of A atoms (analytical balance- N = m/MW)
 Heat transfer (q)
o Several important thermodynamic quantities are not directly measurable- these include:
 Internal energy
 Entropy
 Enthalpy
 (Free energy)
o In addition, several measurable properties of materials are derived quantities- functions of functions (!).
Today we will introduce examples of tthese and examine methods of measuring the quantities of interest.
Calorimetry: Heat capacities and the measurement of internal energy and enthalpy
Measuring heat capacity and enthalpies of phase transitions

In our earlier discussions of temperature and enthalpies of transformation, we’ve already seen that the heat
capacity can be readily determined by measuring the temperature of a material vs. the heat absorbed by it:
(Eqn 1)
dq 
S 
C p     T 
dT P
T P
(Eqn 2)
dq 
S 
CV     T 
dT V
T V
o E.g. for a constant pressure process, Cp is given by the slope of q vs. T:

Lecture 6 – measuring thermodynamic quantities
3 of 10
2/17/16
3.012 Fundamentals of Materials Science
Fall 2003
solid
liquid
qtotal at constant
pressure = H
qtotal
Cpl = dq/dT
Hm
Cps = dq/dT
-40
-20
0
20
40
T (°C)
o

The discontinuity introduced at phase transitions makes calorimetry extremely useful for identifying
melting and boiling points of materials, or temperatures of other phase transitions (such as a change from
one solid structure to another). This kind of plot also directly provides enthalpies of phase transitions,
such as the enthalpy of melting.
Devices to measure temperature vs. heat uptake as shown above are called calorimeters, and are a standard tool
in materials science & engineering. A simple calorimeter as shown below provides electrical heat to a sample,
and measures its temperature as a function of the heat provided.1 Bomb calorimeters are fabricated with fixed
volume high-strength sample containers that allow constant volume measurements (to obtain C v), while other
calorimeters perform measurements at constant pressure. Generally, these devices are designed to heat slowly
enough to approximate a reversible process.
(Dill and Bromberg)2
o
Calorimeters can measure both heat absorption by a sample and heat released by a sample. A common
modern instrument is the differential scanning calorimeter, which you will make use of in the laboratory
session.
Lecture 6 – measuring thermodynamic quantities
4 of 10
2/17/16
3.012 Fundamentals of Materials Science

Fall 2003
As we’ve already seen, knowledge of the heat capacity allows changes in internal energy and entropy to also be
determined as a function of temperature:
Determination of entropies of transitions


Calorimetric data gives us the enthalpy of phase transitions directly- but what about the entropy of these
transitions? Suppose we perform a constant pressure calorimetry experiment, passing through the melting point
of a sample. The heat released on melting is the enthalpy of melting:
o
But we also know from the reversible process definition of the entropy:
o
Thus we find that the entropy of melting can be directly calculated from the enthalpy of melting and the
melting temperature:
So what does Sm look like for different materials?3 It was found experimentally that Sm is approximately the
same for many different metals- Sm ~ 8-16 J/moleK. This correlation, known as Richard’s rule, is seen in the
experimental data below:
o Thus, in the solid to liquid transition, metals with different crystal structures and atomic sizes all have
similar entropy changes: the change in state of aggregation dominates the behavior of the entropy over
smaller atom-specific differences.
Lecture 6 – measuring thermodynamic quantities
5 of 10
2/17/16
3.012 Fundamentals of Materials Science
Fall 2003
(Gaskell)3
o
Measuring expansion and compression

Materials tend to expand in volume as temperature is increased- generally, increasing temperature causes a
loosening of intramolecular bonds. To quantify such changes, the thermal expansion coefficient () is defined as:

(Eqn 6)
o
o
1 dV 
 
V dT P
It is the fractional change in volume with temperature at constant pressure.
 is proportional to the slope of a plot of volume vs. temperature for a material: 1

Lecture 6 – measuring thermodynamic quantities
6 of 10
2/17/16
3.012 Fundamentals of Materials Science
Fall 2003
liquid
solid
~solid
(Dill and Bromerg)
As shown above, both volume and  typically depend on the phase of the material: solids have
lower volume and  than liquids.
Increasing temperature usually leads to expansion of materials, but increasing pressure tends to compress
materials. The isothermal compressibility (), is used to quantify the response of materials to compressive
pressures:


1 dV 

V dP T
   
(Eqn 7)
o

The compressibility is thus the fractional decrease in volume that occurs with pressure at constant
temperature.
 can be extracted from measurements of the volume as a function of pressure:1

~(-
(Dill and Bromberg)

Volume measurements provide insight into the intra-atomic/intra-molecular forces within materials.1 Looking at
the thermal expansion and compressibility data below, one gathers that the bonding between water molecules is
stronger and more ‘rigid’ than in organic liquids- as the expansion coefficient is much lower and water is much
less compressible.
Lecture 6 – measuring thermodynamic quantities
7 of 10
2/17/16
3.012 Fundamentals of Materials Science
Fall 2003
(Dill and Bromberg)

Lecture 6 – measuring thermodynamic quantities
8 of 10
2/17/16
3.012 Fundamentals of Materials Science
Fall 2003
Maxwell Relations

We’ve already discussed some of the useful properties of state functions like U, S, and H- they are path
independent and integrable. An additional useful characteristic is that state functions must obey the Euler
reciprocal relationship, which is expressed mathematically for a multivariable function f(x,y) as:
 2 f (x, y)  2 f (x, y)

xy
ydx
(Eqn 8)
o

The Euler relationship can be used to identify identities between thermodynamic variables that are not
obvious; these are called the Maxwell relations.
A set of Maxwell relations can be derived for each thermodynamic state function. For example, starting with the
 energy U(S,V,N):
internal
 2U
 2U

SV VS
(Eqn 9)

o
You will recognize the term on the right is related to the thermal expansion coefficient. Thus this Maxwell
relation allows us to make the following connection:
…using this equation, we could determine the change in entropy occurring as pressure is
changed at constant volume:

Thus Maxwell’s relations can help us further derive thermodynamic quantities from measurable
parameters of our materials.
Another example is the Maxwell relation obtained by taking second derivatives of the enthalpy H(S,P,N):


(Eqn 13)
(Eqn 14)


 2H
 2H

 S  P P S
V  T 
    
S P P S
Later we will introduce the Gibbs free energy, which has analogous Maxwell relations that may be derived from it.

Lecture 6 – measuring thermodynamic quantities
9 of 10
2/17/16
3.012 Fundamentals of Materials Science
Fall 2003
References
1.
2.
3.
Dill, K. & Bromberg, S. Molecular Driving Forces (New York, 2002).
Dill, K. & Bromberg, S. Molecular Driving Forces (New York, 2003).
Gaskell, D. R. Introduction to Metallurgical Thermodynamics (Hemisphere, New York, 1981).
Lecture 6 – measuring thermodynamic quantities
10 of 10
2/17/16