## The Empirical Gas Laws: Temperature and Charles' Law

James Richard Fromm

The conventional liquid-in-glass thermometer was invented in the seventeenth century. This bulb-and-tube device is still in use; it is shown in the Figure below. In these thermometers the diameter of the bulb is much greater than the diameter of the tube so that a small change in the volume of liquid in the bulb will produce a large change in the height of the liquid in the tube. Two things were not clear about the thermometer at this time. The first question was what it was that the thermometer measured. As the temperature or "degree of hotness" apparent to one's fingers increased, the height of the liquid obviously did also, and this was useful in medicine for checking fevers, but there was no quantitative measurement made, merely the relative degree of hotness between this and that. The second question was whether the degree of hotness of any particular thing was a constant everywhere so that the temperatures of other things could be measured relative to it. Suggested fixed temperatures included that of boiling water, that of melting butter, and the apparently uniform temperature of deep cellars.

Robert Boyle, whose work on the pressure of gases was discussed in a different section, knew of the thermometer, and also was aware that a gas expands when heated. However, since no quantitative temperature scale then existed he could not, and did not, determine the relationship between degree of hotness (temperature) and volume of a gas quantitatively. Boyle did propose a scale of temperature, suggesting that use of a specific fluid in a standardized thermometer bulb with a capacity of 10,000 units filled at the boiling point of water would give a proper scale if changes were at the one-unit level; that is, one degree would have a volume of 10,001 units. His scale was not adopted.

Guillaume Amontons (d. 1705) developed the air thermometer, which uses the increase in the volume of a gas with temperature rather than the volume of a liquid. The air thermometer is an excellent demonstration of Charles' law because the atmosphere maintains a fixed downward pressure above a small mercury plug of constant mass. The volume of a trapped sample of air increases on heating until the pressure of the trapped air equals the pressure of the atmosphere plus the small pressure due to the plug. Nevertheless, Amontons failed to achieve formulation of Charles' law for the same reason as did Boyle: a quantitative scale of temperature was needed.

A quantitative scale of temperature could only be developed after it was realized that at a fixed pressure any pure substance undergoes a phase change at a single fixed temperature which is characteristic of that substance. (We shall explore this question in other sections.) The melting point of ice to water was taken as 0oC and the boiling point of water was taken as 100oC to give our common Celsius scale of temperature. The measurements of the French chemists used the very similar Reaumur scale (water freezes at 0oRe and boils at 80oRe) to establish the law of Charles.

The study of the effect of temperature upon the properties of gases took considerably longer to achieve a simple quantitative relation than did study of the effect of pressure, primarily because the development of a quantitative scale of temperature was a difficult process. However, once such a scale was developed, the appropriate measurements were made, primarily by the French chemist Jacques Charles (1746 - 1823).

The experimental data were formulated into a general law which became known as the law of Charles or Charles' law:

At any constant pressure, the volume of any sample of any gas is directly proportional to the temperature.

Mathematically, the law of Charles can be expressed as

V = k't + k"

where t represents the temperature on any convenient temperature scale and k' and k" are constants. However, as the graph shows, the volume extrapolates to zero at a temperature of -273.15oC. If this temperature were taken as the zero of a temperature scale, the constant k" would be zero and it could be dropped from the equation. Such a temperature scale is now the fundamental scale of temperature in the SI. It is called the absolute scale, the thermodynamic scale, or the Kelvin scale. Temperature on the Kelvin scale, and only on the Kelvin scale, is symbolized by T. The unit of temperature on the Kelvin scale is called the kelvin, and it has the same size as the degree Celsius. The symbol for the unit kelvin is K.

The law of Charles can be written more simply using the Kelvin scale of temperature as V = k'T, where T represents the absolute temperature. An alternative form, more useful when the volume of one particular sample of gas changes with temperature, is V1/T1 = V2/T2.

Example. The volume of a sample of gas is 23.2 mL at 20oC. If the gas is ideal and the pressure remains unchanged, its volume at 80oC will be given by 23.2 mL(353.15 K/293.15 K) = 27.95 mL.