56
2 Macroscopic Thermodynamics
isotherm (T high )
isotherm (T low )
adiabat
adiabat
(a)
1
2
3
4
V
P
1
2
3
4
isotherm (T high )
isotherm (T low )
adiabat
adiabat
(b)
ln V
ln P
Fig. 2.2 Carnot cycle in the P V -plane for a classical ideal monatomic gas. (a) Traditional
representation. (b) Representation using logarithmic scales on the pressure and volume axes
expressed in terms of a particular pressure–volume point in the P V -plane, as shown
in Fig. 2.2. The most common Carnot cycle consists of an isothermal expansion of
the working fluid from thermodynamic state 1 to thermodynamic state 2, followed
by an adiabatic expansion from state 2 to state 3, an isothermal compression from
state 3 to state 4, and finally, an adiabatic compression to return the working fluid
from state 4 to the initial state 1. Such a cycle, progressing in a clockwise direction
in the P V -plane, is said to move in the forward direction.
We shall consider a classical ideal gas to exemplify this process. As illustrated
in Fig. 2.2, the Carnot cycle is associated with the path in the P V -plane taken
by a working (ideal) gas in passing from an initial point (V 1 , P 1 ) by a reversible
isothermal expansion at temperature T high to the point (V 2 , P 2 ), thence via an
adiabatic expansion, during which the temperature changes from T high to T low ,
from (V 2 , P 2 ) to (V 3 , P 3 ), followed by a reversible isothermal compression at
temperature T low from (V 3 , P 3 ) to (V 4 , P 4 ) and, finally, via an adiabatic compression
in which the temperature is taken from T low back to T high and returned to the initial
thermodynamic state (V 1 , P 1 ).
The first step in the Carnot cycle involves the work
W 1→2 (gas) = −
V 2
V 1
P dV = −
V 2
V 1
Nk B T high
V
dV
= −Nk B T high ln
V 2
V 1
< 0 , V 2 > V 1 ,
done on the gas. Because U system (T , V ) ≡ U ideal gas (T ), U 1→2 (gas) = 0 for an
isothermal change, and hence Q 1→2 (gas) = −W 1→2 (gas), so that
2 Macroscopic Thermodynamics
isotherm (T high )
isotherm (T low )
adiabat
adiabat
(a)
1
2
3
4
V
P
1
2
3
4
isotherm (T high )
isotherm (T low )
adiabat
adiabat
(b)
ln V
ln P
Fig. 2.2 Carnot cycle in the P V -plane for a classical ideal monatomic gas. (a) Traditional
representation. (b) Representation using logarithmic scales on the pressure and volume axes
expressed in terms of a particular pressure–volume point in the P V -plane, as shown
in Fig. 2.2. The most common Carnot cycle consists of an isothermal expansion of
the working fluid from thermodynamic state 1 to thermodynamic state 2, followed
by an adiabatic expansion from state 2 to state 3, an isothermal compression from
state 3 to state 4, and finally, an adiabatic compression to return the working fluid
from state 4 to the initial state 1. Such a cycle, progressing in a clockwise direction
in the P V -plane, is said to move in the forward direction.
We shall consider a classical ideal gas to exemplify this process. As illustrated
in Fig. 2.2, the Carnot cycle is associated with the path in the P V -plane taken
by a working (ideal) gas in passing from an initial point (V 1 , P 1 ) by a reversible
isothermal expansion at temperature T high to the point (V 2 , P 2 ), thence via an
adiabatic expansion, during which the temperature changes from T high to T low ,
from (V 2 , P 2 ) to (V 3 , P 3 ), followed by a reversible isothermal compression at
temperature T low from (V 3 , P 3 ) to (V 4 , P 4 ) and, finally, via an adiabatic compression
in which the temperature is taken from T low back to T high and returned to the initial
thermodynamic state (V 1 , P 1 ).
The first step in the Carnot cycle involves the work
W 1→2 (gas) = −
V 2
V 1
P dV = −
V 2
V 1
Nk B T high
V
dV
= −Nk B T high ln
V 2
V 1
< 0 , V 2 > V 1 ,
done on the gas. Because U system (T , V ) ≡ U ideal gas (T ), U 1→2 (gas) = 0 for an
isothermal change, and hence Q 1→2 (gas) = −W 1→2 (gas), so that