235 



For the pressure p is found : 



p = -U -\ (19) 



For helium at normal density ^„^0.239. Hence formula (17j 



would be valid for helium at normal density at extremely low 



temperatures only. At these low temperatures helium at that density 



can no longer exist as an ideal gas. If we call zero point pressure, 



p^, the value which p according to (19) assumes for ^=0, p^ is a 



quantity which immediately enables us to get an estimate of the 



deviations from the Charles's law which are to be expected at low 



temperatures. For helium at normal density the zero point pressure 



is found to be ^) 332 baryes = 0.25 mm. If from this we deduced 



3 

 (he temperature according to p^RT/v, a temperature — <9„=:0°.09 



would correspond to it. The error in the reading of such a thermo- 

 meter, which would occur according to the above application of the 

 quantum-theory, will remain below this amount. 



For the purpose of the theory of free electrons in metals we 

 write (19) in the form '') 



p = aV-'U ^ bT' V, (20) 



where a and h are constants whose values can be easily dei-ived. 

 It is easily verified that the tirst term of (20) does not cause 

 any decrease of temperature at adiabatic expansion: external work 

 is done at T=0 at the expense of the zero point energy. 



§ 5. The specific heat. a. From (4) the specific heat at constant 

 volume can be derived : for this purpose it is to be taken into 

 consideration that r,„ depends according to (5) and (7) on the tem- 

 perature'). We will write down only a few terms of the two cor- 

 responding developments. 



1) The result obtained here differs from that obtained by Sackur I.e., although 

 the above deductions in many respects run parallel to his. 



-) The occurrence of the positive power of V gives a warning that the range 

 of validity of this formula at larger V extends to correspondingly lower T only. 



3) An expression is found, which can be brought to the following form : 



a,^ 5 yc^jsoi rc^\ , 1 ./"^ + ^ 



a.jsoi 10 e^ — l 



For the values of ( -^^ ) , i.e. the ratio of the specific heat for a solid belonging 



\Cao Jsol 



to X to its limiting value at high temperatures see Debue I.e. p. 803, or Nernft, 

 Berlin Sitz.-Ber. 5 Dec. 191i2. 



