Page 58 - Physical chemistry eng

2.10 CALCULATING q, w, ¢U , AND ¢H FOR PROCESSES INVOLVING IDEAL GASES 35

problems. The first law states that ¢U = q + w . Imagine that the process is carried
out under constant volume conditions and that nonexpansion work is not possible.
Because under these conditions w =- P external dV = , 0
1
¢U = q V (2.25)

Equation (2.25) states that ¢U can be experimentally determined by measuring the
heat flow between the system and surroundings in a constant volume process.
Chemical reactions are generally carried out under constant pressure rather than
constant volume conditions. It would be useful to have an energy state function that has
a relationship analogous to Equation (2.25), but at constant pressure conditions. Under
constant pressure conditions, we can write

dU = dq - P external dV = dq - PdV (2.26)
P
P
Integrating this expression between the initial and final states:
f
dU = U - U = dq - PdV = q - P(V - V )
i
f
P
P
f
i
3 L L
i
= q - AP V - P V B (2.27)
i i
f f
P
Note that in order to evaluate the integral involving P, we must know the functional
relationship P(V), which in this case is P = P = P where P is constant. Rearranging
i
f
the last equation, we obtain
(U + P V ) - (U + P V ) = q P (2.28)
i
i i
f
f f
Because P, V, and U are all state functions, U + PV is a state function. This new state
function is called enthalpy and is given the symbol H.
H K U + PV (2.29)
As is the case for U, H has the units of energy, and it is an extensive property. As shown
in Equation (2.30), ¢H for a process involving only P–V work can be determined by
measuring the heat flow between the system and surroundings at constant pressure:
¢H = q P (2.30)
This equation is the constant pressure analogue of Equation (2.25). Because chemical
reactions are much more frequently carried out at constant P than constant V, the
energy change measured experimentally by monitoring the heat flow is ¢H rather than
¢U . When we classify a reaction as being exothermic or endothermic, we are talking
about ¢H , not ¢U .

Calculating q, w, ≤U , and ≤H for
2.10 Processes Involving Ideal Gases

In this section we discuss how ¢U and ¢H , as well as q and w, can be calculated from
the initial and final state variables if the path between the initial and final state is
known. The problems at the end of this chapter ask you to calculate q, w, ¢U , and ¢H
for simple and multistep processes. Because an equation of state is often needed to
carry out such calculations, the system will generally be an ideal gas. Using an ideal
gas as a surrogate for more complex systems has the significant advantage that the
mathematics is simplified, allowing one to concentrate on the process rather than the
manipulation of equations and the evaluation of integrals.
What does one need to know to calculate ¢U ? The following discussion is
restricted to processes that do not involve chemical reactions or changes in phase.
Because U is a state function, ¢U is independent of the path between the initial and
final states. To describe a fixed amount of an ideal gas (i.e., n is constant), the values of
two of the variables P, V, and T must be known. Is this also true for ¢U for processes
involving ideal gases? To answer this question, consider the expansion of an ideal gas