539
REVIEW & SUM MARY
Thermal Expansion All objects change size with changes in temperature. For a temperature change ⌬T, a change ⌬L in any linear
dimension L is given by
⌬L ϭ La ⌬T,
(18-9)
in which a is the coefficient of linear expansion. The change ⌬V in
the volume V of a solid or liquid is
⌬V ϭ Vb ⌬T.
(18-10)
Here b ϭ 3a is the material’s coefficient of volume expansion.
Heat Heat Q is energy that is transferred between a system and
its environment because of a temperature difference between
them. It can be measured in joules (J), calories (cal), kilocalories
(Cal or kcal), or British thermal units (Btu), with
1 cal ϭ 3.968 ϫ 10
Ϫ3
Btu ϭ 4.1868 J.
(18-12)
Heat Capacity and Specific Heat If heat Q is absorbed by
an object, the object’s temperature change T f Ϫ T i is related to Q by
Q ϭ C(T f Ϫ T i ),
(18-13)
in which C is the heat capacity of the object. If the object has mass
m, then
Q ϭ cm(T f Ϫ T i ),
(18-14)
where c is the specific heat of the material making up the object.
The molar specific heat of a material is the heat capacity
per mole, which means per 6.02 ϫ 10
23
elementary units of the
material.
Heat of Transformation Matter can exist in three common
states: solid, liquid, and vapor. Heat absorbed by a material may
change the material’s physical state — for example, from solid to liquid or from liquid to gas.The amount of energy required per unit mass
to change the state (but not the temperature) of a particular material
is its heat of transformation L.Thus,
Q ϭ Lm.
( 1 8 - 1 6 )
The heat of vaporization L V is the amount of energy per unit mass
that must be added to vaporize a liquid or that must be removed to
condense a gas. The heat of fusion L F is the amount of energy per
unit mass that must be added to melt a solid or that must be removed to freeze a liquid.
Work Associated with Volume Change A gas may
exchange energy with its surroundings through work. The amount
of work W done by a gas as it expands or contracts from an initial
volume V i to a final volume V f is given by
(18-25)
W ϭ ͵dW ϭ ͵
V f
V i
p dV.
The integration is necessary because the pressure p may vary during the volume change.
First Law of Thermodynamics The principle of conservation of energy for a thermodynamic process is expressed in the
first law of thermodynamics, which may assume either of the
forms
⌬E int ϭ E int, f Ϫ E int,i ϭ Q Ϫ W
(first law)
(18-26)
or
dE int ϭ dQ Ϫ dW
(first law).
(18-27)
E int represents the internal energy of the material, which depends
only on the material’s state (temperature, pressure, and volume).
Q represents the energy exchanged as heat between the system
and its surroundings; Q is positive if the system absorbs heat and
negative if the system loses heat. W is the work done by the system; W is positive if the system expands against an external force
from the surroundings and negative if the system contracts because of an external force. Q and W are path dependent; ⌬E int is
path independent.
Applications of the First Law The first law of thermodynamics finds application in several special cases:
adiabatic processes: Q ϭ 0, ⌬E int ϭ ϪW
constant-volume processes: W ϭ 0, ⌬E int ϭ Q
cyclical processes: ⌬E int ϭ 0, Q ϭ W
free expansions: Q ϭ W ϭ ⌬E int ϭ 0
Conduction, Convection, and Radiation The rate P cond at
which energy is conducted through a slab for which one face is
maintained at the higher temperature T H and the other face is
maintained at the lower temperature T C is
(18-32)
Here each face of the slab has area A, the length of the slab (the
distance between the faces) is L, and k is the thermal conductivity
of the material.
Convection occurs when temperature differences cause an energy transfer by motion within a fluid.
Radiation is an energy transfer via the emission of electromagnetic energy. The rate P rad at which an object emits energy via thermal radiation is
P rad ϭ s´AT
4
,
(18-38)
where s (ϭ 5.6704 ϫ 10
Ϫ8
W/m
2 иK
4
) is the Stefan – Boltzmann
constant, ´ is the emissivity of the object’s surface, A is its surface
area, and T is its surface temperature (in kelvins). The rate P abs at
which an object absorbs energy via thermal radiation from its environment, which is at the uniform temperature T env (in kelvins), is
P abs ϭ s´AT
4
env .
(18-39)
P cond ϭ
Q
t
ϭ kA
T H Ϫ T C
L
REVIEW & SUM MARY
Thermal Expansion All objects change size with changes in temperature. For a temperature change ⌬T, a change ⌬L in any linear
dimension L is given by
⌬L ϭ La ⌬T,
(18-9)
in which a is the coefficient of linear expansion. The change ⌬V in
the volume V of a solid or liquid is
⌬V ϭ Vb ⌬T.
(18-10)
Here b ϭ 3a is the material’s coefficient of volume expansion.
Heat Heat Q is energy that is transferred between a system and
its environment because of a temperature difference between
them. It can be measured in joules (J), calories (cal), kilocalories
(Cal or kcal), or British thermal units (Btu), with
1 cal ϭ 3.968 ϫ 10
Ϫ3
Btu ϭ 4.1868 J.
(18-12)
Heat Capacity and Specific Heat If heat Q is absorbed by
an object, the object’s temperature change T f Ϫ T i is related to Q by
Q ϭ C(T f Ϫ T i ),
(18-13)
in which C is the heat capacity of the object. If the object has mass
m, then
Q ϭ cm(T f Ϫ T i ),
(18-14)
where c is the specific heat of the material making up the object.
The molar specific heat of a material is the heat capacity
per mole, which means per 6.02 ϫ 10
23
elementary units of the
material.
Heat of Transformation Matter can exist in three common
states: solid, liquid, and vapor. Heat absorbed by a material may
change the material’s physical state — for example, from solid to liquid or from liquid to gas.The amount of energy required per unit mass
to change the state (but not the temperature) of a particular material
is its heat of transformation L.Thus,
Q ϭ Lm.
( 1 8 - 1 6 )
The heat of vaporization L V is the amount of energy per unit mass
that must be added to vaporize a liquid or that must be removed to
condense a gas. The heat of fusion L F is the amount of energy per
unit mass that must be added to melt a solid or that must be removed to freeze a liquid.
Work Associated with Volume Change A gas may
exchange energy with its surroundings through work. The amount
of work W done by a gas as it expands or contracts from an initial
volume V i to a final volume V f is given by
(18-25)
W ϭ ͵dW ϭ ͵
V f
V i
p dV.
The integration is necessary because the pressure p may vary during the volume change.
First Law of Thermodynamics The principle of conservation of energy for a thermodynamic process is expressed in the
first law of thermodynamics, which may assume either of the
forms
⌬E int ϭ E int, f Ϫ E int,i ϭ Q Ϫ W
(first law)
(18-26)
or
dE int ϭ dQ Ϫ dW
(first law).
(18-27)
E int represents the internal energy of the material, which depends
only on the material’s state (temperature, pressure, and volume).
Q represents the energy exchanged as heat between the system
and its surroundings; Q is positive if the system absorbs heat and
negative if the system loses heat. W is the work done by the system; W is positive if the system expands against an external force
from the surroundings and negative if the system contracts because of an external force. Q and W are path dependent; ⌬E int is
path independent.
Applications of the First Law The first law of thermodynamics finds application in several special cases:
adiabatic processes: Q ϭ 0, ⌬E int ϭ ϪW
constant-volume processes: W ϭ 0, ⌬E int ϭ Q
cyclical processes: ⌬E int ϭ 0, Q ϭ W
free expansions: Q ϭ W ϭ ⌬E int ϭ 0
Conduction, Convection, and Radiation The rate P cond at
which energy is conducted through a slab for which one face is
maintained at the higher temperature T H and the other face is
maintained at the lower temperature T C is
(18-32)
Here each face of the slab has area A, the length of the slab (the
distance between the faces) is L, and k is the thermal conductivity
of the material.
Convection occurs when temperature differences cause an energy transfer by motion within a fluid.
Radiation is an energy transfer via the emission of electromagnetic energy. The rate P rad at which an object emits energy via thermal radiation is
P rad ϭ s´AT
4
,
(18-38)
where s (ϭ 5.6704 ϫ 10
Ϫ8
W/m
2 иK
4
) is the Stefan – Boltzmann
constant, ´ is the emissivity of the object’s surface, A is its surface
area, and T is its surface temperature (in kelvins). The rate P abs at
which an object absorbs energy via thermal radiation from its environment, which is at the uniform temperature T env (in kelvins), is
P abs ϭ s´AT
4
env .
(18-39)
P cond ϭ
Q
t
ϭ kA
T H Ϫ T C
L