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## Main Question or Discussion Point

Hei

I need to know the ratio of specific heats, [tex]\gamma[/tex] for an ultra-relativistic gas, in which kT >> [tex]m_{p}c^{2}[/tex], assuming that it is satisfied the equation for a politropic gas [tex]\epsilon=\frac{P}{\gamma-1}[/tex], where [tex]\epsilon[/tex] is the internal energy density.

(What is the difference between relativistic and ultra-relativistic?)

It must be something very easy, I have already the solution for:

[tex]\epsilon=\frac{3}{2}nkT+\frac{3}{2}nkT[/tex]

[tex]P = nkT + nkT[/tex]

So [tex]\gamma=5/3[/tex].

[tex]\epsilon=\frac{3}{2}nkT+3nkT[/tex]

[tex]P = nkT + \frac{1}{3}3nkT[/tex]

So [tex]\gamma=13/9[/tex].

But all this doesn't make much sense to me, could you shed some light over it, please?

I need to know the ratio of specific heats, [tex]\gamma[/tex] for an ultra-relativistic gas, in which kT >> [tex]m_{p}c^{2}[/tex], assuming that it is satisfied the equation for a politropic gas [tex]\epsilon=\frac{P}{\gamma-1}[/tex], where [tex]\epsilon[/tex] is the internal energy density.

(What is the difference between relativistic and ultra-relativistic?)

It must be something very easy, I have already the solution for:

**Ionized Non-relativistic gas:**(kT<< [tex]m_{e}c^{2}[/tex])[tex]\epsilon=\frac{3}{2}nkT+\frac{3}{2}nkT[/tex]

[tex]P = nkT + nkT[/tex]

So [tex]\gamma=5/3[/tex].

**Ionized Relativistic gas:**([tex]m_{e}c^{2}[/tex] << kT << [tex]m_{p}c^{2}[/tex])[tex]\epsilon=\frac{3}{2}nkT+3nkT[/tex]

[tex]P = nkT + \frac{1}{3}3nkT[/tex]

So [tex]\gamma=13/9[/tex].

But all this doesn't make much sense to me, could you shed some light over it, please?