Solids and Liquids
There are approximate equations of state for gases which can give virtually any degree of accuracy desired. However, there are no analogous equations of state for solids and liquids. Fortunately the volumes of solids and liquids do not change very much with pressure as long as the pressure changes are not too large. This situation allows us to define parameters and form an approximate equation of state which is valid over a moderate range of temperatures and pressures.
We will restrict our attention to isotropic liquids and solids, which means that we are excluding liquid crystals and solid single crystals. Single crystals and liquid crystals are anisotropic. Their response to pressure and their expansion with temperature is different along different axes in the crystal. (Many solids, particularly metals and alloys are conglomerates of microscopic crystals with random orientations so that the bulk material behaves like an isotropic solid even though the individual microscopic crystals are anisotropic. We can apply our methods for isotropic substances to these materials even though, strictly speaking, they are crystalline.)
The volume of a sample of an isotropic material is known experimentally to be a function of temperature and pressure. Therefore, we can write,
(1)
(The volume is also a function of the number of moles in the sample, but we will be looking at relative changes, or fractional changes, so that the quantity of material will cancel out.)
We write a differential change in the volume due to differential changes in the temperature and/or the pressure as follows:
(2)
The relative change, or fractional change, is then,
(3)
The coefficients of dp and dT in Equation (3) are so important that we give then names and special symbols.
(4)
is called the isothermal compressibility. The subscript, T, on the lower case Greek letter kappa is to distinguish this compressibility from another related one which will be defined later. When the pressure is increased the volume decreases so that the derivative in Equation (4) is negative. The negative sign in the definition of kT ensures that kappa is positive When there is no concern about confusion we will omit the subscript on the kappa.
(5)
is called the coefficient of thermal expansion (or sometimes just the expansion coefficient).
Values of a and kT must be obtained from experimental data and they can be found in data tables. a and kT are themselves functions of temperature and pressure although they vary so slowly with temperature and pressure that they may usually be regarded as constants except over very large temperature or pressure intervals. We will regard them as constant.
Although a and kT are most useful for liquids and solids, they can be calculated for gases. The volume of a gas is a strong function of temperature and pressure so a and kT are not even approximately constant for gases. It is a useful exercise in the application of partial derivatives to calculate these quantities for an ideal gas. For example, using the ideal gas equation of state we get,
so that
(6,a,b,c)
This quantity is clearly not constant. (Bear in mind that this is the coefficient of thermal expansion for an ideal gas, not the general expression for a. We will leave it to the reader to show that the isothermal compressibility for an ideal gas is 1/p)
Equation (3) can be rewritten, using a and kT as
(7)
which can be integrated to give an approximate equation of state for isotropic liquids and solids,
where Vo is the volume at po and To. It is a useful exercise for the reader to show that this approximate equation of state is consistent with our definitions of a and kT .
There is one other quantity of interest which can be obtained from a and kT, namely,
This is the derivative that tells us how fast the pressure rises when we try to keep the volume constant while increasing the temperature. Using a variation of Euler's chain rule we can write,
Let's apply this to see how much pressure would be generated in a mercury thermometer if we tried to heat the thermometer higher than the temperature where the mercury has reached the top of the thermometer.. For Hg, a = 1.82 ´ 10-4 K-1 and kT = 3.87 ´ 10-5 atm-1. We write.
So we see that each 1 oC increases the pressure by 4.7 atm, about 113 lb/sq in. This is a lot of pressure for a glass tube to withstand. It wouldn't take very many degrees of temperature increase to break the glass thermometer.
Thermometers and the Ideal Gas Temperature Scale
Many of the thermometers we see and use are made of a thin glass tube containing a liquid. The temperature is measured by observing how far up the tube the liquid rises. However, we have already seen that a is not a constant so that liquid expansion is not uniform and the rise in the liquid is not linear with temperature. Worse, different liquids have different nonlinear expansions.
We could pick a standard substance and all agree to measure temperature by the expansion of this substance, but it is unsatisfactory to have our measuring devices tied to particular substances. It would be best if we had a temperature measuring device which was independent of any particular material.
The ideal gas thermometer is such a device and the temperature scale it defines is called the ideal gas temperature scale. The ideal gas temperature scale is based on the fact that all gases become ideal in the limit of zero pressure. Therefore, we can define the ideal gas temperature as,
This temperature scale is independent of the gas used. It has a natural zero since p ³ 0 and V 0, so that pV is never negative. The value of R determines the size of the degree. If R is the gas constant, 0.082057833 Latm/Kmol, then the degree is the Kelvin degree. No one claims that the ideal gas thermometer is easy to use, but it does provide us with an unambiguous theoretical standard to establish a temperature scale.