17.3 Expressing Equilibria with Pressure Terms: Relation Between K
c
and K
p
549
17.3 EXPRESSING EQUILIBRIA WITH PRESSURE TERMS:
RELATION BETWEEN K
c
AND K
p
It is easier to measure the pressure of a gas than its concentration and, as long as
the gas behaves nearly ideally under the conditions of the experiment, the ideal
gas law (Section 5.3) allows us to relate these variables to each other:
PV nRT,so or
where Pis the pressure of a gas and nVis its molar concentration (M). Thus, at
constant temperature, pressure is directly proportional to molar concentration.
When the substances involved in the reaction are gases, we can express the reac-
tion quotient and calculate its value in terms of partial pressures instead of con-
centrations. For example, in the reaction between gaseous NO and O
2
,
2NO(g) O
2
(g) 2NO
2
(g)
the reaction quotient based on partial pressures, Q
p
, is
The equilibrium constant obtained when all components are present at their equi-
librium partial pressures is designated K
p
, the equilibrium constant based on pres-
sures. In many cases, K
p
has a value different from K
c
, but if you know one, you
can calculate the other by noting the change in amount (mol) of gas, n
gas
, from
the balanced equation. Let’s see this relationship by converting the terms in Q
c
for the reaction of NO and O
2
to those in Q
p
:
2NO(g) O
2
(g) 2NO
2
(g)
As the balanced equation shows,
3 mol (2 mol 1 mol) gaseous reactants 2 mol gaseous products
With meaning final minus initial (products minus reactants), we have
n
gas
moles of gaseous product moles of gaseous reactant 2 3 1
Keep this value of n
gas
in mind because it appears in the algebraic conversion
that follows. The reaction quotient based on concentrations is
Rearranging the ideal gas law to nV PRT, we express concentrations as nV
and convert them to partial pressures, P;then we collect the RT terms and cancel:
The far right side of the previous expression is Q
p
multiplied by RT: Q
c
Q
p
(RT).
Also, at equilibrium, K
c
K
p
(RT); thus, K
p
, or K
c
(RT)
1
.
Notice that the exponent of the RT term equals the change in the amount
(mol) of gas (n
gas
) from the balanced equation, 1. Thus, in general, we have
(17.8)
The units for the partial pressure terms in K
p
are generally atmospheres, pas-
cals, or torr, raised to some power, and the units of Rmust be consistent with
those units. As Equation 17.8 shows, for those reactions in which the amount
(mol) of gas does not change, we have n
gas
0, so the RT term drops out
and K
p
K
c
.
K
p
K
c
(RT)
n
gas
K
c
RT
Q
c
n
2
NO
2
V
2
n
2
NO
V
2
n
O
2
V
P
2
NO
2
(RT)
2
P
2
NO
(RT)
2
P
O
2
RT
P
2
NO
2
P
2
NO
P
O
2
1
(RT)
2
1
(RT)
2
1
RT
P
2
NO
2
P
2
NO
P
O
2
RT
Q
c
[NO
2
]
2
[NO]
2
[O
2
]
BA
BA
Q
p
P
2
NO
2
P
2
NO
P
O
2
BA
P
RT n
V
P n
VRT
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