The Molar Specific Heats of an Ideal Gas
In this module, we want to derive from molecular considerations an expression for
the internal energy E int of an ideal gas. In other words, we want an expression for the
energy associated with the random motions of the atoms or molecules in the gas.We
shall then use that expression to derive the molar specific heats of an ideal gas.
Internal Energy E int
Let us first assume that our ideal gas is a monatomic gas (individual atoms rather
than molecules), such as helium, neon, or argon. Let us also assume that the internal energy E int is the sum of the translational kinetic energies of the atoms.
(Quantum theory disallows rotational kinetic energy for individual atoms.)
The average translational kinetic energy of a single atom depends only on the
gas temperature and is given by Eq. 19-24 as
. A sample of n moles of
such a gas contains nN A atoms.The internal energy E int of the sample is then
(19-37)
E int ϭ (nN A )K avg ϭ (nN A )(
3
2 kT ).
K avg ϭ
3
2 kT
564
CHAPTE R 19 TH E KI N ETIC TH EORY OF GASES
19-7 THE MOLAR SPECIFIC HEATS OF AN IDEAL GAS
After reading this module, you should be able to . . .
19.28 Identify that the internal energy of an ideal monatomic gas is
the sum of the translational kinetic energies of its atoms.
19.29 Apply the relationship between the internal energy E int
of a monatomic ideal gas, the number of moles n, and the
gas temperature T.
19.30 Distinguish between monatomic, diatomic, and
polyatomic ideal gases.
19.31 For monatomic, diatomic, and polyatomic ideal gases,
evaluate the molar specific heats for a constant-volume
process and a constant-pressure process.
19.32 Calculate a molar specific heat at constant pressure C p by
adding R to the molar specific heat at constant volume C V ,
and explain why (physically) C p is greater.
19.33 Identify that the energy transferred to an ideal gas as
heat in a constant-volume process goes entirely into the
internal energy (the random translational motion) but that
in a constant-pressure process energy also goes into the
work done to expand the gas.
19.34 Identify that for a given change in temperature, the
change in the internal energy of an ideal gas is the same
for any process and is most easily calculated by assuming
a constant-volume process.
19.35 For an ideal gas, apply the relationship between heat
Q, number of moles n, and temperature change ⌬T, using
the appropriate molar specific heat.
19.36 Between two isotherms on a p-V diagram, sketch a
constant-volume process and a constant-pressure
process, and for each identify the work done in terms of
area on the graph.
19.37 Calculate the work done by an ideal gas for a constantpressure process.
19.38 Identify that work is zero for constant volume.
Learning Objectives
● The molar specific heat C V of a gas at constant volume is
defined as
in which Q is the energy transferred as heat to or from a
sample of n moles of the gas, ⌬T is the resulting temperature
change of the gas, and ⌬E int is the resulting change in the
internal energy of the gas.
● For an ideal monatomic gas,
● The molar specific heat C p of a gas at constant pressure is
C V ϭ
3
2 R ϭ 12.5 J/molиK.
C V ϭ
Q
n ⌬T
ϭ
⌬E int
n ⌬T
,
defined to be
in which Q, n, and
are defined as above. C p is also given by
C p ϭ C V ϩ R.
● For n moles of an ideal gas,
E int ϭ nC V T (ideal gas).
● If n moles of a confined ideal gas undergo a temperature
change ⌬T due to any process, the change in the internal
energy of the gas is
⌬E int ϭ nC V ⌬T (ideal gas, any process).
⌬T
C p ϭ
Q
n ⌬T
,
Key Ideas
In this module, we want to derive from molecular considerations an expression for
the internal energy E int of an ideal gas. In other words, we want an expression for the
energy associated with the random motions of the atoms or molecules in the gas.We
shall then use that expression to derive the molar specific heats of an ideal gas.
Internal Energy E int
Let us first assume that our ideal gas is a monatomic gas (individual atoms rather
than molecules), such as helium, neon, or argon. Let us also assume that the internal energy E int is the sum of the translational kinetic energies of the atoms.
(Quantum theory disallows rotational kinetic energy for individual atoms.)
The average translational kinetic energy of a single atom depends only on the
gas temperature and is given by Eq. 19-24 as
. A sample of n moles of
such a gas contains nN A atoms.The internal energy E int of the sample is then
(19-37)
E int ϭ (nN A )K avg ϭ (nN A )(
3
2 kT ).
K avg ϭ
3
2 kT
564
CHAPTE R 19 TH E KI N ETIC TH EORY OF GASES
19-7 THE MOLAR SPECIFIC HEATS OF AN IDEAL GAS
After reading this module, you should be able to . . .
19.28 Identify that the internal energy of an ideal monatomic gas is
the sum of the translational kinetic energies of its atoms.
19.29 Apply the relationship between the internal energy E int
of a monatomic ideal gas, the number of moles n, and the
gas temperature T.
19.30 Distinguish between monatomic, diatomic, and
polyatomic ideal gases.
19.31 For monatomic, diatomic, and polyatomic ideal gases,
evaluate the molar specific heats for a constant-volume
process and a constant-pressure process.
19.32 Calculate a molar specific heat at constant pressure C p by
adding R to the molar specific heat at constant volume C V ,
and explain why (physically) C p is greater.
19.33 Identify that the energy transferred to an ideal gas as
heat in a constant-volume process goes entirely into the
internal energy (the random translational motion) but that
in a constant-pressure process energy also goes into the
work done to expand the gas.
19.34 Identify that for a given change in temperature, the
change in the internal energy of an ideal gas is the same
for any process and is most easily calculated by assuming
a constant-volume process.
19.35 For an ideal gas, apply the relationship between heat
Q, number of moles n, and temperature change ⌬T, using
the appropriate molar specific heat.
19.36 Between two isotherms on a p-V diagram, sketch a
constant-volume process and a constant-pressure
process, and for each identify the work done in terms of
area on the graph.
19.37 Calculate the work done by an ideal gas for a constantpressure process.
19.38 Identify that work is zero for constant volume.
Learning Objectives
● The molar specific heat C V of a gas at constant volume is
defined as
in which Q is the energy transferred as heat to or from a
sample of n moles of the gas, ⌬T is the resulting temperature
change of the gas, and ⌬E int is the resulting change in the
internal energy of the gas.
● For an ideal monatomic gas,
● The molar specific heat C p of a gas at constant pressure is
C V ϭ
3
2 R ϭ 12.5 J/molиK.
C V ϭ
Q
n ⌬T
ϭ
⌬E int
n ⌬T
,
defined to be
in which Q, n, and
are defined as above. C p is also given by
C p ϭ C V ϩ R.
● For n moles of an ideal gas,
E int ϭ nC V T (ideal gas).
● If n moles of a confined ideal gas undergo a temperature
change ⌬T due to any process, the change in the internal
energy of the gas is
⌬E int ϭ nC V ⌬T (ideal gas, any process).
⌬T
C p ϭ
Q
n ⌬T
,
Key Ideas