Study Guide for Material in Engel & Reid
Chapters 1-4
Fall 2005
I
Chapter 1. Fundamental Concepts of Thermodynamics
A
Systems and Surroundings
It is convenient to separate parts of a thermodynamic problem into system
and surroundings. We can then establish conventions to specify that work
done on a system is positive and heat moving into a system is positive. The
system-surroundings boundary is important for thermodynamic treatments
of heat and work.
The nature of the boundary defines three types of idealized systems.
Isolated systems are surrounded by a boundary that does not allow for the
exchange of either matter or energy with the surroundings, a closed system
has a boundary that allows for heat exchange, but no matter exchange, and
an open system has a boundary that allows matter exchange but no energy
exchange with the surroundings. Boundary walls that do not allow heatenergy exchange are called adiabatic or insulating, while those that do allow
heat-energy exchange are called diathermal.
B
Temperature vs. Heat
Heat is a type of energy in transit – the quantity of energy that flows across
the boundary between systems and surroundings because of a temperature
differences. The Maxwell-Boltzmann distribution relates particle velocities
to a parameter identified as temperature. As heat energy is increased, the
distribution changes such that the average particle velocity increases, which
in turn is said to indicate an increase in temperature.
The zeroth law of thermodynamics helps us define temperature and thermometry. The law states that two systems in thermal equilibrium with a
third system are in thermal equilibrium with each other. There is no net
heat flow between systems in thermal equilibrium, and we say they are at
the same temperature.
1
C
The Ideal Gas Law and Partial Pressures
Much of thermodynamics begins by considering an idealized system of pointmass particles that interact only through elastic collisions (like infinitely
small billiard balls). The equation of state for the system is the ideal gas
law, P V = nRT .
For mixtures of ideal gases, the partial pressures of individual gases are
convenient functions of mole fractions and the total pressure. For a two-gas
(A and B) mixture, for example, the following equations are true for the
ideal case (and Ptotal = PA + PB for any case).
PA = xA Ptotal
PB = xB Ptotal
xA =
nA
ntotal
xA + x B = 1
(1)
Ideal gases are also convenient for certain problems because their energy
(U ) depends only on temperature. For real gases, pressure and volume
would alter the importance of molecular interactions, but for ideal gases (no
molecular interactions), only temperature changes will affect U .
D
The Van der Waals Equation of State
The Van der Waals equation of state improves on the ideal gas law by
accounting for the attractive and repulsive forces of real molecules in a simplified way. The equation of state (shown below) includes a term with a
parameter a (always positive) that accounts for attractive forces between
molecules and therefore reduces the pressure compared to that of the ideal
model. The b-term is also always positive, and as a measure of molecular
volume, it accounts for repulsive interactions between molecules that will
increase pressure compared to the ideal model.
P =
nRT
n2 a
− 2
V − nb
V
(2)
Fits of experimental P-V-T data to the Van der Waals model have been
used to establish tables of a and b parameters for many gases.
II
A
Chapter 2: Heat, Work, Internal Energy, Enthalpy, and the First Law of Thermodynamics
Internal Energy and the First-Law
Many years of experience has suggested to scientists that something is conserved as systems change. Or as Feynman put it, “There is something
2
quantitative in the universe that is conserved and we call it energy”. The
conservation of energy is the essence of the first law, and our focus is typically on the energy of a system – the internal energy, U . The first law
could be stated formally in at least two ways (again, U with no subscript is
Usystem ):
∆U + ∆Usurroundings = 0 dU = d¯q + d¯w
(3)
The differential expression of the first law is particularly useful because it
distinguishes between heat (q) and work (w) coming from the surroundings
to alter the internal energy of a system. Note that dU is an exact differential
(so U is a state function and ∆U = Uf inal − Uinitial ), and d¯q and d¯w are
inexact differentials (so values of q and w will depend on the path, or type
of change in a process).
B
P-V Work
Many problems in chemical thermodynamics include changes in pressure
and/or volume. We showed that P-V changes have the units of energy
(like work) and correspond to classical notions of work as force through a
distance. The differential form for P-V work,
dw = −Pext dV
(4)
is defined in terms of the external pressure P ext applied to the system. For
reversible processes (involving a series of infinitesimal changes such that the
system and surroundings always appear to be at equilibrium), the external
pressure will equal the system pressure, so P ext can be replaced with P (the
system pressure), which may be conveniently expressed using an equation
of state like the ideal gas law.
C
Enthalpy
We saw that for constant volume problems, ∆U was convenient because
it could be equated with q since no P-V work is possible (and assuming
there is no other type of work). For constant pressure problems (with only
PV work), we showed that the state function enthalpy (H = U + P V ) was
similarly useful, because for this class of problems ∆H = q.
D
Heat Capacity
The heat capacity of a substance is a measure of how much heat is required
to change it’s temperature. We used two types of heat capacities defined for
3
constant volume or constant pressure problems.
∂U
∂H
CV =
CP =
∂T V
∂T P
(5)
Heat capacities can be useful because they have been measured for many
substances and they related directly to thermodynamic state functions (U or
H). Also, for ideal gases, there is a simple relationship that can be derived:
CP = CV + nR
III
Chapter 3: The Importance of State Functions: Internal Energy and Enthalpy
Chapter 3 showed how a few rules concerning partial derivatives could be
useful in deriving thermodynamic relationships. The test for an exact differential (a test for a state function) was expressed in terms of the order of differentiation in second derivatives of a two-variable function (say
F = F (x, y)):
!
∂ ∂F
∂ ∂F
=
if dF is exact
(6)
∂x ∂y x y
∂y ∂x y
x
We learned that U could be defined in terms of P ,V , and T for a system,
and since P, V, T could be related through an equation of state, only two
variable were truly needed. Therefore,
∂U
∂U
∂U
dU =
dT +
dV = CV dT +
dV
(7)
∂T V
∂V T
∂V T
∂U
This equation was particularly useful for ideal-gas problems because ∂V
=
T
0 (U for ideal gases depends only on T )
For real systems, we saw that useful thermodynamic information could
be obtained from several experimental parameters.
∂T
µJ−T =
Joule-Thompson Coefficient
(8)
∂P H
1 ∂V
κ=−
Isothermal Compressibility
(9)
V ∂P T
1 ∂V
β=
Volumetric Thermal Expansion Coefficient
(10)
V ∂T P
4
IV
Chapter 4: Thermochemistry
We showed for chemical reactions, the enthalpy change could be expressed
in terms of enthalpies of formation that are commonly found in tables of
thermodynamic data.
X
◦
◦
∆Hreaction
=
νi ∆Hf,i
νi is negative for reactants, positive for products
i
(11)
The standard state formation enthalpies are the enthalpy changes for forming compounds from the elements in their most stable form at the standardstate temperature and pressure (at 298.15 K and 1 bar for example, the
element oxygen would be gaseous O2 .)
We also saw that reaction enthalpies could be estimated from bond enthalpies. The bond enthalpies found in standard tables correspond to the
enthalpy change for homolytic cleavage of a bond into atoms, so to use them
as we used formation enthalpies above, it is necessary to change their sign
(so that they refer to forming a bond from atoms).
Finally, we learned that reaction enthalpies calculated as described above
for the standard state temperature could be adjusted to other temperatures
using heat capacities of reactants and products:
Z T
X
◦
∆CP dT ∆CP =
νi CP,i -reactants, +products
+
∆HT◦ = ∆H298.15
298.15
i
(12)
5