350
9 Bioenergetics
W 12 =
pdV = nRT C ln
V 1
V 2
= Q C
(9.53)
and the work gained from the isothermal process state 3 to state 4 is
W 12 =
pdV = nRT H ln
V 4
V 3
= Q H
(9.54)
The two isothermals are connected by adiabats (i.e., no heat exchange). For these
adiabatic processes,
dE = −pdV
(9.55)
or
c v dT = −
RT
V
dV .
(9.56)
This means that during an adiabatic process,
T V
γ −1
= constant
(9.57)
where
γ ≡
c v + R
c v
(9.58)
With the adiabats connecting the isothermals, we have
T C V
γ −1
2
= T H V
γ −1
3
(9.59)
T C V
γ −1
1
= T H V
γ −1
4
(9.60)
In ratio it follows that
V 2
V 1
=
V 3
V 4
(9.61)
Thus,
Q C
Q H
=
T C
T H
,
(9.62)
showing that the efficiency of an ideal-gas Carnot engine is
9 Bioenergetics
W 12 =
pdV = nRT C ln
V 1
V 2
= Q C
(9.53)
and the work gained from the isothermal process state 3 to state 4 is
W 12 =
pdV = nRT H ln
V 4
V 3
= Q H
(9.54)
The two isothermals are connected by adiabats (i.e., no heat exchange). For these
adiabatic processes,
dE = −pdV
(9.55)
or
c v dT = −
RT
V
dV .
(9.56)
This means that during an adiabatic process,
T V
γ −1
= constant
(9.57)
where
γ ≡
c v + R
c v
(9.58)
With the adiabats connecting the isothermals, we have
T C V
γ −1
2
= T H V
γ −1
3
(9.59)
T C V
γ −1
1
= T H V
γ −1
4
(9.60)
In ratio it follows that
V 2
V 1
=
V 3
V 4
(9.61)
Thus,
Q C
Q H
=
T C
T H
,
(9.62)
showing that the efficiency of an ideal-gas Carnot engine is