2.2 An Introduction to Thermodynamics
35
If the thermodynamic system is contained within adiabatic walls (which do not
permit energy transfer through them), then Q ≡ 0, and U ≡ U f − U i = W , in
which W is the work done on the thermodynamic system. Note that the change U
depends only upon the initial and final thermodynamic states of the system and is
independent of the path taken for the process. Because U is path-independent,
an equivalent reversible path can often be found between the initial and final
thermodynamic states should the equation of state for the system be known. This
is a rather important observation, as the computation of Q and W can be carried out
explicitly for such a reversible path, and their sum provides the value of U for the
actual process, whether or not the process is itself reversible.
When we apply the differential expression (2.2.4a) for the first law to a
thermodynamic system in which only pressure–volume work δW = −P ext dV is
possible, we obtain dU as
dU = δQ − P ext dV .
(2.2.6)
A closed simple chemical system (having, in general, a fixed mass and composition
for which surfaces and external fields are unimportant) satisfies this requirement. If
the process that we wish to consider is adiabatic (i.e., δQ ≡ 0), then Eq. (2.2.6)
gives
dU = −P dV ,
(2.2.7a)
with the pressure P expressible as a function of V via the equation of state should
the process happen also to be reversible. For an irreversible adiabatic process,
however, Eq. (2.2.6) gives simply
dU = −P ext dV ,
(2.2.7b)
for which P ext cannot in general be expressed as a function of V using the equation
of state.
If we consider a reversible isothermal expansion of an ideal gas, the work done
is given by δW = −P dV , while from Eq. (2.2.2b), U = 0 for an isothermal
process (dT ≡ 0), so that Eq. (2.2.4a) for the first law gives the heat transferred as
δQ = P dV = Nk B T d ln V ,
(2.2.8a)
as the pressure P is given as P = Nk B T /V from the ideal gas equation of state.
Integration of Eq. (2.2.8a) from initial state i (with volume V i ) to final state f (with
volume V f ) gives
f
i
δQ
T
= Nk B
V f
V i
d ln V = Nk B ln
V f
V i
.
35
If the thermodynamic system is contained within adiabatic walls (which do not
permit energy transfer through them), then Q ≡ 0, and U ≡ U f − U i = W , in
which W is the work done on the thermodynamic system. Note that the change U
depends only upon the initial and final thermodynamic states of the system and is
independent of the path taken for the process. Because U is path-independent,
an equivalent reversible path can often be found between the initial and final
thermodynamic states should the equation of state for the system be known. This
is a rather important observation, as the computation of Q and W can be carried out
explicitly for such a reversible path, and their sum provides the value of U for the
actual process, whether or not the process is itself reversible.
When we apply the differential expression (2.2.4a) for the first law to a
thermodynamic system in which only pressure–volume work δW = −P ext dV is
possible, we obtain dU as
dU = δQ − P ext dV .
(2.2.6)
A closed simple chemical system (having, in general, a fixed mass and composition
for which surfaces and external fields are unimportant) satisfies this requirement. If
the process that we wish to consider is adiabatic (i.e., δQ ≡ 0), then Eq. (2.2.6)
gives
dU = −P dV ,
(2.2.7a)
with the pressure P expressible as a function of V via the equation of state should
the process happen also to be reversible. For an irreversible adiabatic process,
however, Eq. (2.2.6) gives simply
dU = −P ext dV ,
(2.2.7b)
for which P ext cannot in general be expressed as a function of V using the equation
of state.
If we consider a reversible isothermal expansion of an ideal gas, the work done
is given by δW = −P dV , while from Eq. (2.2.2b), U = 0 for an isothermal
process (dT ≡ 0), so that Eq. (2.2.4a) for the first law gives the heat transferred as
δQ = P dV = Nk B T d ln V ,
(2.2.8a)
as the pressure P is given as P = Nk B T /V from the ideal gas equation of state.
Integration of Eq. (2.2.8a) from initial state i (with volume V i ) to final state f (with
volume V f ) gives
f
i
δQ
T
= Nk B
V f
V i
d ln V = Nk B ln
V f
V i
.