The Equipartition Law in a System of Particles. 595 

 The Relativity Expression for Temperatures. 



We are now in a position to derive the relativity expression 

 for temperature. The thermodynamic scale of temperature 

 may be defined in terms of the efficiency of a heat engine. 

 Consider a four-step cycle performed with a working sub- 

 stance contained in a cylinder provided with a piston. In 

 the first step let the substance expand isothermally and 

 reversibly, absorbing the heat Q 2 from a reservoir at tempe- 

 rature T 2 ; in the second step cool the cylinder down at 

 constant volume to T x ; in the third step compress to the 

 original volume, giving out the heat Q x at temperature T lr 

 and in the fourth step heat to the original temperature. 

 Now if the working substance is of such a nature that the 

 heat given out in the second step could be used for the 

 reversible heating of the cylinder in the fourth step, we may 

 define the absolute temperatures T 2 and T l by the relation 



•1-2 *%2 * 



Consider now such a cycle performed on a cylinder which 

 contains one of our systems of particles. Since we have 

 shown (p. 593) that at a definite temperature the energy 

 content of such a system is independent of the volume, it is 

 evident that our working substance fulfils the requirement 

 that the heat given out in the second step shall be sufficient 

 for the reversible heating in the last step. Hence, in 

 accordance with the thermodynamic scale, we may measure 

 the temperatures of the two heat reservoirs by the relation 



To 



_? = ^ and may proceed to obtain expressions for Q 2 



l-i Wi 

 and Q,. 



In order to obtain these expressions we may again make 

 use of the principle that the energy content at a definite 

 temperature is independent of the volume. This being true. 

 we see that Q 2 and Q! must be equal to the work done in the 

 changes of volume that take place respectively at T 2 and Tj, 

 and we may write the relations 



Q 2 = f ^V(atT 2 ), 

 Qi= pdY{sitT 1 ). 



* We have used this cycle for defining the thermodynamic temperature 

 scale instead of the familiar Carnot cycle, since it avoids the necessitv of 

 knowledge as to adiabatic expansions. 



•1 Q 2 



