THE COMBINED GAS LAW

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

Boyle’s law

Constant

Constant

temperature T 1 temperature T 2 law

Constant pressure P

3 P 2 V new T

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

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

= 2 V 2 ( a given sample of gas)

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.

760 torr

V 1.50 L

P 1 V 1 P 2 V 2 ( 760 torr)(1.50 L)

P ( 0.950 L)

T 1 T 2 350 K

285 K

P 2 = 977 torr

Standard Conditions

According to the combined gas law, the volume of a given sample of gas can have any value, depending on its temperature and pressure. To compare the quantities of gas present in two different samples, it is useful to adopt

a set of standard conditions of temperature and pressure. By universal agreement, the standard temperature is chosen as 273 K (0 ◦ C), and the standard pressure is chosen as exactly 1 atm (760 torr). Together, these conditions are referred to as standard conditions or as standard temperature and pressure (STP). Many textbooks and instructors find it convenient to use this short notation for this particular temperature and pressure.

EXAMPLE 12.10.

A sample of gas occupies a volume of 1.88 L at 22 ◦ C and 0.979 atm pressure. What is the volume of this sample at STP?

Ans.

P 1 = 0.979 atm

= 2 V 2 ( = 0.979 atm)(1.88 L) =

1.00 atm)V