3.9 Quasi-static Heating and the Adiabatic
Transformation of a Gas
We consider for the cases of constant molar heat capacities c p and c v , some
examples of ideal gas processes.
3.9.1 Isochoric processes
For an internally reversible quasi-static isochoric (constant volume V f = V i ) process,
Eq. (29) becomes
@U
@T
V
dT ¼ dQ
ð29AÞ
Integrating Eq. (29A) with the definition of heat capacity, Eq. (32),
Q i!f ¼ Nc V T f À T i
À
Á
From the thermal equation of state for an ideal gas Eqs. (4) or (5).
p
T
¼ const:
ð38Þ
or
p f
p i
¼
T f
T i
Heating leads to an increase in gas pressure in a constant volume container.
3.9.2 Isobaric processes
Note that this is a prediction of heating in a closed constant volume system.
A student may sometimes mistakenly assume that heating associates generally with
pressure increase, and apply this relation to an open flow system. In that case, the
correct prediction will be that the pressure of an open system is determined by the
overall flow configuration (based on Poisson equation for the pressure) and thus
remains little changed as a result of heating, effect of which is limited to local
region; instead, the specific volume of the fluid increases (corresponding to the
result shown below for constant pressure cases) as a result of local change in
temperature.
52
3 The First Law: The Production of Heat …
Transformation of a Gas
We consider for the cases of constant molar heat capacities c p and c v , some
examples of ideal gas processes.
3.9.1 Isochoric processes
For an internally reversible quasi-static isochoric (constant volume V f = V i ) process,
Eq. (29) becomes
@U
@T
V
dT ¼ dQ
ð29AÞ
Integrating Eq. (29A) with the definition of heat capacity, Eq. (32),
Q i!f ¼ Nc V T f À T i
À
Á
From the thermal equation of state for an ideal gas Eqs. (4) or (5).
p
T
¼ const:
ð38Þ
or
p f
p i
¼
T f
T i
Heating leads to an increase in gas pressure in a constant volume container.
3.9.2 Isobaric processes
Note that this is a prediction of heating in a closed constant volume system.
A student may sometimes mistakenly assume that heating associates generally with
pressure increase, and apply this relation to an open flow system. In that case, the
correct prediction will be that the pressure of an open system is determined by the
overall flow configuration (based on Poisson equation for the pressure) and thus
remains little changed as a result of heating, effect of which is limited to local
region; instead, the specific volume of the fluid increases (corresponding to the
result shown below for constant pressure cases) as a result of local change in
temperature.
52
3 The First Law: The Production of Heat …