Page 45 - Essentials of physical chemistry

Ideal and Real Gas Behavior 7

same units as differences in Kelvin temperature. If a chain rule derivative should occur involving
temperature differences, D K ¼ D C; they are the same size unit! While we are establishing simple

facts, we note that 18Cis defined as 1=100th of the temperature between that of ice water and boiling
water (at 1 atm) as established on the metric (French) scale while the British standardized the
Fahrenheit degree as 1=180th of the same temperature gap but then embellished it with the notion
that colder temperatures are possible below ice water. Using concentrated salt solutions (as in
making homemade ice cream) the original Fahrenheit scale was set to the lowest temperature of
water saturated with salt some 328 lower even though we now know far lower temperatures are
possible. Thus, we have

180 9
C þ 32

F ¼ C þ 32 ¼
100 5
It is easy to show that the two temperature scales give the same value at 40 by substituting one for

the other either way. In Fahrenheit we see that absolute zero is

9
( 273:15 ) þ 32 ¼ 459:67 F:

F ¼
5
The Fahrenheit scale is still used in engineering but in chemistry and physics the centigrade scale is
used along with the absolute temperature K.

Now let us combine Boyle’s law with Charles’ law to improve the overall phenomenological
description (for a fixed mass of gas). Consider a two-step process that first changes the pressure of
the gas followed by a change in the temperature.

P 1 T 1 V 1 ! P 2 T 1 V x ! P 2 T 2 V 2

P 1
V 1 and then use Charles’ law at constant
Then by Boyle’s law at constant temperature V x ¼
pressure as P 2

T 2 P 1 T 2 V 2 P 1 T 2
V 2 ¼ V x ¼ V 1 ) ¼ :
T 1 P 2 T 1 V 1 P 2 T 1
This leads to a very useful equation for a fixed mass of gas as

P 1 V 1 P 2 V 2
¼
T 1 T 2
This equation implies the existence of another constant C 3 .

PV
¼ C 3 ¼ nR
T

Now we are close to a general phenomenological equation that takes into account P, V, T, and the
moles (n) of the gas, which fixes the mass of the gas sample. Usually this equation is given (in high
school chemistry!) as

PV ¼ nRT

So what is R? And now we come to a key development in 1876.