148
3 The Second and Third Laws of Thermodynamics: Entropy
A general procedure for the calculation of an entropy change in a closed system
without phase change or chemical reaction is to carry out the line integral
∆S
c
dq rev
T
over a curve in state space corresponding to a reversible path.
The statistical entropy is defined by
S st k B ln(Ω) + S 0
where k B is Boltzmann’s constant and where Ω is the thermodynamic probability,
equal to the number of mechanical states that might be occupied by the system given
its macroscopic state. The quantity S 0 is an arbitrary constant whose value we can
take to equal zero. We showed that for an isothermal expansion of a model system
the change in the statistical entropy and the change in the thermodynamic entropy are
equal.
The third law of thermodynamics asserts that if the entropies of all samples of pure
perfect crystalline elements are taken as zero, the entropies of all samples of pure perfect
crystalline compounds can also consistently be taken as zero. Entropies relative to the
entropy at zero temperature are called absolute entropies. The values of these absolute
entropies can be used to calculate entropy changes of chemical reactions.
A consequence of the second and third laws of thermodynamics is that no finite
number of operations, either reversible or irreversible, can bring an object to zero
temperature.
A D D I T I O N A L P R O B L E M S
3.45 a. Sketch a diagram representing the Carnot cycle in a
state space with S on the horizontal axis and T on the
vertical axis.
b. Show that the work done on the surroundings in
one cycle is equal to the area enclosed in this
diagram.
3.46 Assume that a substance A can react without any energy
change to an isomer C (as in a racemization
reaction).
a. Assume that 1.000 mol of substance A is placed in a
container that is adiabatically isolated from any
surroundings and is allowed to equilibrate. Write an
expression for the entropy change of the system as a
function of x, the amount of A that has reacted.
b. Find the value of x at which the entropy of the system
is at a maximum.
c. Find the value of the equilibrium constant for the
reaction.
3.47 Supercooled (metastable) water vapor commonly occurs in
the atmosphere if dust particles are not present to begin
condensation to the liquid. Sometimes small particles, such
as tiny crystals of silver iodide, are released from airplanes
in an attempt to begin condensation. This process is called
“cloud seeding.” At a certain location, water vapor at 25 ◦ C
has a metastable partial pressure of 32.0 torr. The equilibrium value at this temperature is 23.756 torr. Consider the
air that is present to be the surroundings, and assume it to
remain at equilibrium at 25 ◦ C. A tiny particle is added to
begin condensation. Calculate ∆S, ∆H, and ∆S surr per
mole of water that condenses. State any assumptions.
3.48 Without doing any detailed calculations, specify for each
process whether each of the following quantities is positive, negative, or equal to zero: q, w, ∆U, ∆S, and ∆S surr .
a. The system consisting of two large objects and a small
bar between them is allowed to come to equilibrium
from an initial state in which the two objects are at
different temperatures. Assume that the objects have
fixed volume.
3 The Second and Third Laws of Thermodynamics: Entropy
A general procedure for the calculation of an entropy change in a closed system
without phase change or chemical reaction is to carry out the line integral
∆S
c
dq rev
T
over a curve in state space corresponding to a reversible path.
The statistical entropy is defined by
S st k B ln(Ω) + S 0
where k B is Boltzmann’s constant and where Ω is the thermodynamic probability,
equal to the number of mechanical states that might be occupied by the system given
its macroscopic state. The quantity S 0 is an arbitrary constant whose value we can
take to equal zero. We showed that for an isothermal expansion of a model system
the change in the statistical entropy and the change in the thermodynamic entropy are
equal.
The third law of thermodynamics asserts that if the entropies of all samples of pure
perfect crystalline elements are taken as zero, the entropies of all samples of pure perfect
crystalline compounds can also consistently be taken as zero. Entropies relative to the
entropy at zero temperature are called absolute entropies. The values of these absolute
entropies can be used to calculate entropy changes of chemical reactions.
A consequence of the second and third laws of thermodynamics is that no finite
number of operations, either reversible or irreversible, can bring an object to zero
temperature.
A D D I T I O N A L P R O B L E M S
3.45 a. Sketch a diagram representing the Carnot cycle in a
state space with S on the horizontal axis and T on the
vertical axis.
b. Show that the work done on the surroundings in
one cycle is equal to the area enclosed in this
diagram.
3.46 Assume that a substance A can react without any energy
change to an isomer C (as in a racemization
reaction).
a. Assume that 1.000 mol of substance A is placed in a
container that is adiabatically isolated from any
surroundings and is allowed to equilibrate. Write an
expression for the entropy change of the system as a
function of x, the amount of A that has reacted.
b. Find the value of x at which the entropy of the system
is at a maximum.
c. Find the value of the equilibrium constant for the
reaction.
3.47 Supercooled (metastable) water vapor commonly occurs in
the atmosphere if dust particles are not present to begin
condensation to the liquid. Sometimes small particles, such
as tiny crystals of silver iodide, are released from airplanes
in an attempt to begin condensation. This process is called
“cloud seeding.” At a certain location, water vapor at 25 ◦ C
has a metastable partial pressure of 32.0 torr. The equilibrium value at this temperature is 23.756 torr. Consider the
air that is present to be the surroundings, and assume it to
remain at equilibrium at 25 ◦ C. A tiny particle is added to
begin condensation. Calculate ∆S, ∆H, and ∆S surr per
mole of water that condenses. State any assumptions.
3.48 Without doing any detailed calculations, specify for each
process whether each of the following quantities is positive, negative, or equal to zero: q, w, ∆U, ∆S, and ∆S surr .
a. The system consisting of two large objects and a small
bar between them is allowed to come to equilibrium
from an initial state in which the two objects are at
different temperatures. Assume that the objects have
fixed volume.