36
2 Macroscopic Thermodynamics
Moreover, as T is held fixed during the process, we may write (δQ)/T = δ
Q
T
,
so that we obtain
Q
T
= Nk B ln
V f
V i
.
(2.2.8b)
Thus, because the temperature is fixed for an isothermal process, the final volume
V f is determined by the amount of energy Q transferred into the system from
the surroundings; V f can thereby be arbitrarily set (despite the fact that both T i
and T f have been fixed). This result, like the restriction (2.2.3) obtained earlier
for an adiabatic process, also extends beyond the ideal gas, thereby implying that
in general the entropy must change, with the consequence that S = 0 for any
reversible isothermal process. Because S changes during an isothermal process, we
are thereby free to choose both V i and V f arbitrarily.
For a general change of state for an ideal structureless gas, we obtain (δQ)/T
from the first law as
δQ
T
= C V d ln T + Nk B d ln V
(2.2.9a)
or in an equivalent integral form as
f
i
δQ
T
= C V
T f
T i
d ln T + Nk B
V f
V i
d ln V
= C V ln
T f
T i
+ Nk B ln
V f
V i
.
(2.2.9b)
As the right-hand side of Eq. (2.2.9b) depends only upon the initial and final
thermodynamic states characterized by the thermodynamic variables (T , V ), the
quantity (δQ)/T on the left-hand side of Eq. (2.2.9b) must represent an exact (total)
differential that is zero for an adiabatic process and gives the result (2.2.8b) for an
isothermal process. This behaviour suggests that T −1 serves as an integrating factor
that then enables us to make the identification
dS =
δQ
T
,
(2.2.10a)
with the consequence that the macroscopic change S is given by
f
i
δQ
T
= S f − S i ≡ S .
(2.2.10b)
The Second Law of Thermodynamics is essentially a statement of the impossibility of attaining 100% efficiency in the conversion between heat and work, and
2 Macroscopic Thermodynamics
Moreover, as T is held fixed during the process, we may write (δQ)/T = δ
Q
T
,
so that we obtain
Q
T
= Nk B ln
V f
V i
.
(2.2.8b)
Thus, because the temperature is fixed for an isothermal process, the final volume
V f is determined by the amount of energy Q transferred into the system from
the surroundings; V f can thereby be arbitrarily set (despite the fact that both T i
and T f have been fixed). This result, like the restriction (2.2.3) obtained earlier
for an adiabatic process, also extends beyond the ideal gas, thereby implying that
in general the entropy must change, with the consequence that S = 0 for any
reversible isothermal process. Because S changes during an isothermal process, we
are thereby free to choose both V i and V f arbitrarily.
For a general change of state for an ideal structureless gas, we obtain (δQ)/T
from the first law as
δQ
T
= C V d ln T + Nk B d ln V
(2.2.9a)
or in an equivalent integral form as
f
i
δQ
T
= C V
T f
T i
d ln T + Nk B
V f
V i
d ln V
= C V ln
T f
T i
+ Nk B ln
V f
V i
.
(2.2.9b)
As the right-hand side of Eq. (2.2.9b) depends only upon the initial and final
thermodynamic states characterized by the thermodynamic variables (T , V ), the
quantity (δQ)/T on the left-hand side of Eq. (2.2.9b) must represent an exact (total)
differential that is zero for an adiabatic process and gives the result (2.2.8b) for an
isothermal process. This behaviour suggests that T −1 serves as an integrating factor
that then enables us to make the identification
dS =
δQ
T
,
(2.2.10a)
with the consequence that the macroscopic change S is given by
f
i
δQ
T
= S f − S i ≡ S .
(2.2.10b)
The Second Law of Thermodynamics is essentially a statement of the impossibility of attaining 100% efficiency in the conversion between heat and work, and