MSE 308
Thermodynamics of Materials
Dept. of Materials Science & Engineering
Spring 2005/Bill Knowlton
Problem Set 1 Solutions
1. In sections1.3, Gaskell states the final volume, V3, is independent of the order in which
the steps to attain V3 are taken. Explain why this is so. Use section 1.2 to help you.
Because the state function is an exact differential and the state
function is given by equation 1.1 and equation 1.4. In thermodynamics, we
only care about comparing the initial state and the final state. We do not
care by what path it changes from initial to final. The whole reason we
can ignore the path is that exact differentials are path independent.
More on this later.
2. In section 1.6, Gaskell gives a brief synopsis of extensive and intensive properties (or
variables) of a system. According to his definitions, determine whether the following
properties (or variables) are extensive or intensive and explain your reasoning.
a. y = y1 = y2 = y3,
where y1, y2 and y3 are variables measured in a system that is in equilibrium.
i. Provide several examples of variables.
Y is an intensive variable because at each point in the system it is the
same as the system as a whole (point variable). Examples are pressure
and temperature.
b. x = x1 + x2 + x3
where x1, x2 and x3 are variables measured in a system that is in equilibrium.
i. Provide several examples of variables.
Y is an extensive variable because at each point in the system it is not
the equal to the system as a whole and if the sum of all Y’s at each point
is the value of the system as a whole (point variable). Examples are
volume and number of particles in the system.
3. Define the following terms:
a. State function
A function that describes the state of a system using variable that can
move the system away from equilibrium. p. 4 Gaskell
b. Equation of state
Very similar to a state function in that it is an equation that describes
the state of a system when given values of variables that can move the
system away from equilibrium. p. 2 Gaskell
1
MSE 308
Thermodynamics of Materials
Dept. of Materials Science & Engineering
Spring 2005/Bill Knowlton
c. Constitution or phase diagram
Typically a phase diagram is a graphical means to represent equilibrium
states. p. 9 Gaskell
d. Process variables
Process variables, such as heat and work, are path dependent and thus
have meaning only for a changing system. The way we use a process
variable in a sentence eludes to the fact that it is path dependent and it’s
value depends on path. For instance, “work done on or by a system” and
“the heat transferred to or from a system” are good examples. Hence,
the differentials of a process variable (dQ and dW) are inexact.
e. Homogeneous system
A homogeneous system is one that has only one phase. (p. 10 Gaskell)
f. Heterogeneous system
A heterogeneous system is one that has more than one phase (p. 10
Gaskell).
g. Unary system
A unary system is one that has one component. (p. 9 Gaskell)
h. Binary system
A binary system is one that has two components. (p. 9 Gaskell)
i. Ternary system
A ternary system is one that has three components. (p. 9 Gaskell)
j. Quaternary system
A Quaternary system is one that has four components. (p. 9 Gaskell)
2
MSE 308
Thermodynamics of Materials
Dept. of Materials Science & Engineering
Spring 2005/Bill Knowlton
4. On page 7 of Gaskell, figure 1.3a, a plot of pressure versus volume of an ideal gas is
plotted at two different temperatures.
a. By examining the plot, what do you think the relationship is between pressure as a
function of volume? In other words, what do you think is the mathematical
functionality? Explain.
The data is decreasing with increasing volume therefore there seems to
be an inverse relationship. Thus:
1
[1]
P∝
V
We also notice that as temperature increases that the pressure
increases so there seems to be a direct relationship between pressure
and temperature. Thus:
P ∝T
[2]
Combining equations [1] & [2], we have
T
P ∝ . [3]
V
We now need a constant of proportionality This gives:
T
[4]
P=R
V
Why use R? If we remember the equation of state for an ideal gas, we
know that:
PV = nRT [5]
Where n is the number of moles, or
PV = RT [6]
for 1 mole of gas. If we solve for pressure, we equation [4]. We can plot
this in Mathematica where R is the ideal gas constant which is 8.314
(J/mol K) = 0.08206 (atm L)/(mol K)). We shall use the latter value for
the plotting.
3
MSE 308
Thermodynamics of Materials
Dept. of Materials Science & Engineering
Spring 2005/Bill Knowlton
b. Using your reasoning from part a, use a mathematical program (not Excel) to plot
figure 1.3a. Label your plot thoroughly.
MSE 308
Bill Knowlton
ProblemSet 1
Problem 4b:
Clear@v, p, r, T, PplotD
r = 0.08206; H∗universal gas constant, units = Hatm LLêHmol KL 8.314 JêHmol KL∗L
rT
p@v_, T_D :=
H∗Ideal Gas Law for 1 mole of gas∗L
v
H∗Plots∗L
Pplot1 = Plot@p@v, 300D, 8v, 1, 20<, PlotRange → 80, 20<, Frame → True,
GridLines → Automatic, PlotStyle → RGBColor@0, 0, 1D,
FrameLabel → 8"Volume HLL", "Pressure HatmL"<,
PlotLabel → "Red =300K; Blue=1000K"D;
Pplot2 = Plot@p@v, 1000D, 8v, 1, 20<, PlotRange → 80, 20<, Frame → True,
GridLines → Automatic, PlotStyle → RGBColor@1, 0, 0D,
FrameLabel → 8"Volume HLL", "Pressure HatmL"<,
PlotLabel → "Red = 300 K; Blue = 1000 K"D;
Show@Pplot1, Pplot2D;
Red =300K; Blue =1000 K
20
Pressure HatmL
17.5
15
12.5
10
7.5
5
2.5
0
5
10
Volume HLL
15
20
Red = 300 K; Blue = 1000 K
20
Pressure HatmL
17.5
15
12.5
10
7.5
5
2.5
0
5
10
Volume HLL
15
20
Red =300K; Blue =1000 K
20
17.5
Pressure HatmL
In[78]:=
15
12.5
10
7.5
5
2.5
0
5
10
Volume HLL
15
4
20