MSE 308
Thermodynamics of Materials
Dept. of Materials Science & Engineering
Spring 2005/Bill Knowlton
Problem Set 2 Solutions
1. Prove that the inexact differential, dw is equal to mechanical work, pdV:
i.e, dw = pdV
when the change in all other work is constant. Use the fact that
dw = Fdx
where F is force and x is distance.
i = Fdx
dw
F
= dx ⋅ A
A
= PdV
where
F
A
and
V = x⋅ A
P=
MSE 308
Thermodynamics of Materials
Dept. of Materials Science & Engineering
Spring 2005/Bill Knowlton
2. Prove that mechanical work on a system is negative. Use a drawing of a cylinder and
piston to prove this and the concept of state 1 with variable V1 and P1 and state 2 with
variables V2 and P2.
MSE 308
Thermodynamics of Materials
Dept. of Materials Science & Engineering
Spring 2005/Bill Knowlton
3. The equation I provided in class that describes the 1st Law of Thermodynamics has an
opposite sign as equation (2.2) provided by Gaskell on page 19. Explain why there is a
difference.
The equation (2.2) is given by:
dU = dQ − dw .
The negative sign denote that work is being done by the system rather than
on the system which is positive. With this convention, W is defined to be
positive when it is transmitted from the system to the surroundings; thus, if
W is positive, the internal energy of the system decreases.
4. Contrast the relative magnitudes of the entropy transfer versus entropy production in the
following processes:
a. A thermally insulated container has two compartments of equal size. Initially, one
side is filled with a gas and the other is evacuated. A valve is opened and the gas
expands to fill both compartments.
In the thermally insulated case, there is no entropy transfer; the total
entropy change in this case is entropy production.
b. A gas contained in a steel cylinder is slowly expanded to twice its volume.
Slow expansion minimizes dissipation effects, and thus is accompanied by a
small production of entropy; most of the entropy change in this case is
entropy transfer.
MSE 308
Thermodynamics of Materials
Dept. of Materials Science & Engineering
Spring 2005/Bill Knowlton
5. The notion of a “reversible process is a fiction in the real world. What make this concept,
which at first glance would appear to be only of academic interest, so useful in applying
thermodynamics to real-world “irreversible” process?:
If the property changes that are of interest in a real irreversible process
are state functions, then their changes will be the same for every process
that connects the initial and final states of the process. In particular, these
changes will be the same as for some reversible process that takes the
system between these two states. Calculation of changes for a reversible
process is very much simpler than for irreversible processes because
internal intensive properties like temperature and pressure are uniform in
the system. Further, for reversible processes, general relationships are
available for computing W and Q from state function information by
integrating:
dQrev = TdS
δWrev = -PdV
along the simplest path that connects the initial and final states. Thus,
calculations of changes in state functions for irreversible processes are
made accessible through the useful fiction of reversible processes.
MSE 308
Thermodynamics of Materials
Dept. of Materials Science & Engineering
Spring 2005/Bill Knowlton
6. Prove that the entropy in the universe is always increasing. Use the method we described
in class using the change of entropy for the universe, surroundings and system.