Paul Ander
Seton Hall University south Orange, New Jersey 07079
Concept of Empirical Temperature for Introductory Chemistry
W h i l e the term temperature is used throughout the introductory chemistry course, I have never seen a simple development of the concept of empirical temperature in an introductory chemistry text. The following presentation' is used in my freshman course prior to the discussion of the empirical gas laws for dilute gases, i.e., Boyle's, Charles's, Dalton's, etc. Temperature is a property of such paramount importance that it should be understood both from an experimental point of view as well as from its theoretical interpretation. While temperature is easily understood when interpreted from the simple kinetic theory of dilute gases, this interpretation does not aid us to use temperature as an experimental observable. Hence, the concept of empirical temperature must be developed. The idea of temperature is linked to the common physiological experience of hotness and coldness. According to our senses we even endeavor to establish a crude scale by introducing such words as warm and cool between the extremes of hot and cold. However, such a scale would not be useful scientifically inasmuch zs it would depend upon the observer, and it could not be given a value on a suitable, reproducible scale. What is desired is an operational definition of temperature which clearly defines it as a physical property so that it could he measured in the laboratory objectively. To develop an operational definition of temperature, a terminology must be introduced to make our discussion precise. A system is defined as any portion of the universe one chooses to study. I t could he a hook or the solar system. The system employed by many chemists is simply chemicals in a flask. For our purpose here, we consider only macroscopic systems, i.e., those systems whose physical properties can be measured by suitable laboratory instruments. Consider the system of a definite mass of gas enclosed in a container whose walls are non-rigid. (This system will he employed because its simplicity facilitates the discussion.) To describe this system macroscopically, a minimum number of experimental variables must be specified for a complete description. When the numerical values of this minimum number of experimental variables are specified, they are collectively referred to as the state of the system. To describe the system macroscopically we can measure the mechanical propl D s ~ n ~ rK., c ~ "The , Principles of Chemical Equilibrium," Cambridge University Press, London, 1955, Chapter 1. Itom, P. A,, "Chemical Thermodynamics," iMacmillan, New York, 1969 Chapter 1. REDLICH, O., J. CHEM.EDUC.,47, 740 (1970).
erties of mass, volume, and pressure of the gas. However, if these properties are measured on a hot day and on a cold day, the mass will remain the same, but the volume and/or the pressure will change. So stating the mass, volume, and pressure of an amount of gas is not sufficient to describe the state of the gas. Then, to completely specify the state of this system, a new variable, a thermal variable, must he introduced. It is called temperature, the concept of which will he developed. When two or more gaseous systems are brought into contact along their diathermal walls t o permit heat exchange, the observable properties of each system will change until the observable properties of each system attain constant values. Then the systems are in thermal equilibrium. The observable variables for a gaseous system, mass, pressure, and volume of the gas, will be used to define a thermal variable as an observable property of the system. Consider two gaseous systems A and B each with a fixed mass of gas. System A is a t pressure and volume PA,VAand system B is at pressure and volume PB, V B . NOWbring the systems into contact along their diathermal walls so that thermal energy may flow between the systems. The pressure and volumes of each system change. When thermal equilibrium is reached the pressure and volume of each system achieve the constant values PA',VA' for system A and PB1,VB'for system B. Separate system A from system B. System B with its thermal equilibrium state PB',VBtwill be used as a reference to see if system A is at the same thermal equilibrium. Now readjust the pressure and volume of system A to find a new set of values for these variables such that system A is still in thermal equilibrium with system B a t PBf,Va'. After testing for this experimentally, it is found that system A a t PA",VAf'is in thermal equilibrium with system B at PBJ,Ve'. Logically then system A at PA",VA"would be in thermal equilibrium with system A a t PA1,VA'. The Zeroth Law of Thermodynamics states that when two systems are in thermal equilibrium with a third system, they are in thermal equilibrium with one another.= Again readjust the pressure and volume of system A to find a new set of values for these variables such that system A is still in thermal equilibrium with system B at PBf,V.'. Experimentally i t is found that system A at PA"',VA"' is in thermal equilibrium with system B at PBf,Vn'. Repeat this procedure several times and plot the results. These are illustrated in the figure. Each state of system A shown in the figure is in thermal equilibrium with one another and with system B at state P,',VBf. A line connecting these states is called an isotherm. The thermal variable common to these states is called temperature. So temperature is that property of a system Volume 48, Number 5, May
1971
/
325
the length of liquid column can be used as the thennometric property. Since on heating, the volume expansion of liquids is much larger than the linear expansion of metals, the length of a liquid column finds greater use as a thermometric property. So to decide on the property to be used, the temperature variation of various properties is examined and the decision is made from consideration of the magnitude of the property change per degree, of the simplicity of the relationship, and of the convenience of the measurement. Any functional relationship will do be it a quadratic, a cubic, or a square root relationship, etc. For convenience it is best to choose the simplest of relationships a linear relationship, between the thermometric property X and the empirical temperature t t=a+bX
I
v The stater of a flxed moss of gos of system A in thermal equilibrium with one another.
which determines whether it will remain in thermal equilibrium on thermal contact (to allow heat flow) with other systems. (It is this basic quantity temperature in addition to the four independent basic quantities of mechanics and electricity: mass, length, time, and electric current, which give rise to the subject of thermodynamics.) Next a scale must be established to relate the thermal variable temperature to measurable properties so that the thermal variable can he quantified. The relationship betveen the temperature and the thermometric property must be single-valued and monotonic. This will ensure that one thermometric property measurement will give only one temperature value. Thus the volume of a sample of water between 0' and 4'C is a poor thermometric property since it is neither singlevalued or monotonic in this range. Now it must be decided on what the values of the thermometric property should be to correspond to a value of the empirical temperature. Since many physical properties change with a change in temperature, a large choice of physical properties are avai1:~ble for us to use as the thermometric property. Also the thermometric property should be one that is easily and precisely measurable. Most metals and liquids expand when heated, so the length of metallic strip and
326
/
Journol o f Chemical Educofion
(1)
where a and b are two empirical constants needed to establish the equation of the line. Since both a and b must he determined, two conditions or reference points must be defined. The reference points are defined for convenience. One is the ice point which is an equilibrium mixture of ice and water at 1 atm pressure. The second is the steam point which is an equilibrium mixture of water and steam a t 1 atm pressure. The thermometric property at the ice point is X,.. and for convenience, a t x,.. the value given to t is zero. The thermometric property a t the steam point is X,t.., and, for convenience, at X,*, the value given to t is 100. So a t the ice point eqn. (1) becomes 0 =a
+ bXi.,
('4
and a t the steam point eqn. (1) becomes 100 = a
+ bX.e.m
(3)
Solving eqns. (1) and (2) simultaneously for a and b. and inserting the values for a and b into eqn. (1) gives t =
X - Xi,, . 100 X 8 k , - X
