Page 191 - Theory and Problems of BEGINNING CHEMISTRY

180 GASES [CHAP. 12

12.6. THE COMBINED GAS LAW
Suppose it is desired to calculate the final volume V 2 of a gas originally at volume V 1 when its temperature
is changed from T 1 to T 2 at the same time its pressure is changed from P 1 to P 2 . One might consider the two
effects separately, for example, that first the pressure is changed at constant temperature T 1 and calculate a new
volume V new using Boyle’s law. Then, using Charles’ law, one can calculate how the new volume V new changes
to V 2 when the temperature is changed from T 1 to T 2 at the constant pressure P 2 (boxes 1, 3, and 4 in Fig. 12-8).
It would be equally correct to consider that first the temperature of the gas was changed from T 1 to T 2 at
the constant pressure P 1 , for which a new volume V new could be calculated using Charles’ law. Then, assuming
that the temperature is held constant at T 2 , calculate how the volume changes as the pressure is changed from P 1
to P 2 (boxes 1, 2, and 4 in Fig. 12-8).

Constant pressure P 1
1 P 1 V 1 T 1 P 1 V new T 2 2
Charles’ law

Boyle’s Constant Constant Boyle’s
law temperature T 1 temperature T 2 law

Constant pressure P 2
3 P 2 V new T 1 P 2 V 2 T 2 4
Charles’ law
Fig. 12-8. Change in gas volume with both pressure and temperature

However, the fact that the volume V of a given mass of gas is inversely proportional to its pressure P and
directly proportional to its absolute temperature T can be combined mathematically to give the single equation

T
V = k
P
where k is the proportionality constant. Rearranging the variables gives the following equation:
PV
= k
T
That is, for a given sample of gas, the ratio PV/T remains constant, and therefore

P 1 V 1 P 2 V 2
= (a given sample of gas)
T 1 T 2
This expression is a mathematical statement of the combined (or general) gas law. In words, the volume of a
given sample of gas is inversely proportional to its pressure and directly proportional to its absolute temperature.
Note that if the temperature is constant, T 1 = T 2 , then the expression reduces to the equation for Boyle’s
law, P 1 V 1 = P 2 V 2 . Alternatively, if the pressure is constant, P 1 = P 2 , the expression is equivalent to Charles’
law V 1 /T 1 = V 2 /T 2 .

EXAMPLE 12.9. A sample of gas is pumped from a 1.50-L vessel at 77 C and 760-torr pressure to a 0.950-L vessel at
◦
12 C. What is its final pressure?
◦
Ans.
1 2
P 760 torr P 2
V 1.50 L 0.950 L
T 77 C = 350 K 12 C = 285 K
◦
◦
P 1 V 1 P 2 V 2 (760 torr)(1.50 L) P 2 (0.950 L)
= = =
T 1 T 2 350 K 285 K
P 2 = 977 torr