Page 26 - Physical Chemistry

lev38627_ch01.qxd 2/20/08 11:38 AM Page 7

7
C to be in thermal equilibrium with each other. By our definition of temperature, we Section 1.3
would assign the same temperature to A and B (u u ) and the same temperature to Temperature
B
A
B and C (u u ). Therefore, systems A and C would have the same temperature
B
C
(u u ), and we would expect to find A and C in thermal equilibrium when they
C
A
are brought in contact via a thermally conducting wall. If A and C were not found to
be in thermal equilibrium with each other, then our definition of temperature would be
invalid. It is an experimental fact that:
Two systems that are each found to be in thermal equilibrium with a third sys-
tem will be found to be in thermal equilibrium with each other.
This generalization from experience is the zeroth law of thermodynamics. It is so called
because only after the first, second, and third laws of thermodynamics had been for-
mulated was it realized that the zeroth law is needed for the development of thermody-
namics. Moreover, a statement of the zeroth law logically precedes the other three. The
zeroth law allows us to assert the existence of temperature as a state function.
Having defined temperature, how do we measure it? Of course, you are familiar
with the process of putting a liquid-mercury thermometer in contact with a system,
waiting until the volume change of the mercury has ceased (indicating that thermal
equilibrium between the thermometer and the system has been reached), and reading
the thermometer scale. Let us analyze what is being done here.
To set up a temperature scale, we pick a reference system r, which we call the
thermometer. For simplicity, we choose r to be homogeneous with a fixed composi-
tion and a fixed pressure. Furthermore, we require that the substance of the ther-
mometer must always expand when heated. This requirement ensures that at fixed
pressure the volume of the thermometer r will define the state of system r uniquely—
two states of r with different volumes at fixed pressure will not be in thermal equilib-
rium and must be assigned different temperatures. Liquid water is unsuitable for a
thermometer since when heated at 1 atm, it contracts at temperatures below 4°C and
expands above 4°C (Fig. 1.5). Water at 1 atm and 3°C has the same volume as water
at 1 atm and 5°C, so the volume of water cannot be used to measure temperature.
Liquid mercury always expands when heated, so let us choose a fixed amount of liquid
mercury at 1 atm pressure as our thermometer.
We now assign a different numerical value of the temperature u to each different
volume V of the thermometer r. The way we do this is arbitrary. The simplest
r
approach is to take u as a linear function of V . We therefore define the temperature to
r
be u aV b, where V is the volume of a fixed amount of liquid mercury at 1 atm
r r Figure 1.5
pressure and a and b are constants, with a being positive (so that states which are ex-
perienced physiologically as being hotter will have larger u values). Once a and b are Volume of 1 g of water at 1 atm
specified, a measurement of the thermometer’s volume V gives its temperature u. versus temperature. Below 0°C,
r the water is supercooled (Sec. 7.4).
The mercury for our thermometer is placed in a glass container that consists of a
bulb connected to a narrow tube. Let the cross-sectional area of the tube be A, and let
the mercury rise to a length l in the tube. The mercury volume equals the sum of the
mercury volumes in the bulb and the tube, so
u aV b a1V bulb Al2 b aAl 1aV bulb b2 cl d (1.3)
r
where c and d are constants defined as c aA and d aV b.
bulb
To fix c and d, we define the temperature of equilibrium between pure ice and liq-
uid water saturated with dissolved air at 1 atm pressure as 0°C (for centigrade), and
we define the temperature of equilibrium between pure liquid water and water vapor
at 1 atm pressure (the normal boiling point of water) as 100°C. These points are called
the ice point and the steam point. Since our scale is linear with the length of the mer-
cury column, we mark off 100 equal intervals between 0°C and 100°C and extend the
marks above and below these temperatures.