3.4 The pressure of a non-relativistic degenerate gas

In the non-relativistic limit (i.e. when v << c), the total energy of the gas may be expressed as

cap e sub NR equals cap n sub e times left parenthesis three times p sub cap f squared divided by 10 times m sub e plus m sub e times c squared right parenthesis

Equation label:(9)

The second term in the brackets in Equation (9) is simply the rest-mass energy per particle, and the first term is the kinetic energy per particle. Furthermore, the kinetic energy per unit volume is the kinetic energy per particle multiplied by the number of particles per unit volume. So the kinetic energy per unit volume is three times p sub cap f squared times n sub e solidus left parenthesis 10 times m sub e right parenthesis .

Now, the pressure provided by non-relativistic particles is 2/3 of the kinetic energy per unit volume, so

equation sequence part 1 cap p sub NR equals part 2 two divided by three multiplication three times p sub cap f squared times n sub e divided by 10 times m sub e equals part 3 n sub e times p sub cap f squared divided by five times m sub e full stop

Then, using Equation (8), we have

equation sequence part 1 cap p sub NR equals part 2 n sub e divided by five times m sub e multiplication left parenthesis three times n sub e divided by eight times pi right parenthesis super two solidus three times h squared equals part 3 h squared divided by five times m sub e times left parenthesis three divided by eight times pi right parenthesis super two solidus three times n sub e super five solidus three full stop

The equation of state for non-relativistic degenerate electrons may therefore be written as follows:

cap p sub NR equals cap k sub NR times n sub e super five solidus three where cap k sub NR equals h squared divided by five times m sub e times left parenthesis three divided by eight times pi right parenthesis super two solidus three full stop

Equation label:(10)

OpenLearn - White dwarfs and neutron stars
ou logo Except for third party materials and otherwise, this content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 Licence, full copyright detail can be found in the acknowledgements section. Please see full copyright statement for details.