Page 52 - Physical chemistry eng

2.6 STATE FUNCTIONS AND PATH FUNCTIONS 29

an intermediate state in which the volume is V using a constant external pressure
2
P external where V 6 V 1 . The work is given by Mass
2
Mechanical
V f V f
Piston
stops
w =- P external dV =-P external dV =-P external (V - V ) =-P external ¢V (2.15)
i
f
3 3
V i V i
P 1 ,V ,T 1
1
Because work is done on the system in the compression (see Figure 2.11), w is positive
and U increases. Because the system consists of a uniform single phase, U is a monoto-
nic function of T, and T also increases. The change in volume ¢V has been chosen such
that the temperature of the system T in the intermediate state after the compression sat-
2
isfies the inequality T 6 T 6 T 3 .
1
2
We next lock the piston in place and let an amount of heat q flow between the system T 3
and surroundings at constant V by bringing the system into contact with the reservoir at
temperature T . The final state values of T and V after these two steps are T and V . Initial state
3
3
2
This two-step process is repeated for different values of the external pressure by
changing the mass resting on the piston. In each case the system is in the same final state
characterized by the variables V and T . The sequence of steps that takes the system
2
3
from the initial state V ,T to the final state V ,T is referred to as a path. By changing Mass
2
3
1
1
the mass, a set of different paths is generated, all of which originate from the state V ,T , Piston
1
1
and end in the state V ,T . According to the first law, ¢U for this two-step process is
2
3
P ,V ,T
¢U = U(T ,V ) - U(T ,V ) = q + w (2.16) 2 2 2
2
3
1
1
Because ¢U is a state function, its value for the two-step process just described is the
same for each of the different values of the mass. T 3
Are q and w also state functions? For this two step process,
w =-P external ¢V (2.17) Intermediate state
and P external is different for each value of the mass or for each path, whereas ¢V is
constant. Therefore, w is also different for each path; we can choose one path from
V ,T to V ,T and a different path from V ,T back to V ,T . Because the work is dif- Mass
2
1
3
1
3
2
1
1
ferent along these paths, the cyclic integral of work is not equal to zero. Therefore, w
is not a state function. Piston
Using the first law to calculate q for each of the paths, we obtain the result
P ,V ,T 3
3
2
q =¢U - w =¢U + P external ¢V (2.18)
Because ¢U is the same for each path, and w is different for each path, we conclude that
q is also different for each path. Just as for work, the cyclic integral of heat is not equal to T 3
zero. Therefore, neither q nor w are state functions, and they are called path functions.
Because both q and w are path functions, there are no exact differentials for work and Final state
heat unless the path is specified. Incremental amounts of these quantities are denoted by
dq and dw , rather than dq and dw, to emphasize the fact that incremental amounts of FIGURE 2.11
work and heat are not exact differentials. Because dq and dw are not exact differentials, A system consisting of an ideal gas is
there are no such quantities as ¢q , q , q and ¢w , w , w . One cannot refer to the work or contained in a piston and cylinder
i
f
f
i
heat possessed by a system or to the change in work or heat associated with a process. assembly. An external pressure is
After a transfer of heat and/or work between the system and surroundings is completed, generated by the mass resting on the
the system and surroundings possess internal energy, but not heat or work. piston.The gas in the initial state V 1 ,T 1 is
The preceding discussion emphasizes that it is important to use the terms work and compressed to an intermediate state,
whereby the temperature increases to the
heat in a way that reflects the fact that they are not state functions. Examples of systems of
value T 2 . It is then brought into contact
interest to us are batteries, fuel cells, refrigerators, and internal combustion engines. In each
with a thermal reservoir at T 3 , leading to a
case, the utility of these systems is that work and/or heat flows between the system and sur-
further rise in temperature. The final state
roundings. For example, in a refrigerator, electrical energy is used to extract heat from the
is V 2 ,T 3 . The mechanical stops allow the
inside of the device and to release it in the surroundings. One can speak of the refrigerator as system volume to be only V 1 or V 2 .
having the capacity to extract heat, but it would be wrong to speak of it as having heat. In
the internal combustion engine, chemical energy contained in the bonds of the fuel mole-
cules and in O 2 is released in forming CO 2 and H O . This change in internal energy can be
2
used to rotate the wheels of the vehicle, thereby inducing a flow of work between the vehi-
cle and the surroundings. One can refer to the capability of the engine to do work, but it
would be incorrect to refer to the vehicle or the engine as containing or having work.