- #1
Apashanka
- 429
- 15
From the energy equation E=m0c2/√(1-v2/c2) for non-relativistic gas molecules (v<<c) ,E reduces to m0c2...(1)
From ideal gas law PV=nRT
P=nRT/V
P=nkBNAT/V
P=(nNA)kBT/V
P=(nNAm0)kBT/v
P=mtotalkBT/vm0
P=(mtotal/V)kBT/m0
P=ρkBT/m0
(If n moles of a gas is taken in volume V at temp T and volume V,m0 being the mass of each molecule at equipibrium)
From equipartition theorem
3/2kBT=m0c2(total energy of each molecule)
kBT/m0=2/3c2
Putting this in P becomes
P=2/3ρc2
P=2/3ε(energy density)
Therefore (P=ωε) ω=2/3 (Nonrelativistic gas)
But for matter dominated universe we take ω=0
Why is it so??
From ideal gas law PV=nRT
P=nRT/V
P=nkBNAT/V
P=(nNA)kBT/V
P=(nNAm0)kBT/v
P=mtotalkBT/vm0
P=(mtotal/V)kBT/m0
P=ρkBT/m0
(If n moles of a gas is taken in volume V at temp T and volume V,m0 being the mass of each molecule at equipibrium)
From equipartition theorem
3/2kBT=m0c2(total energy of each molecule)
kBT/m0=2/3c2
Putting this in P becomes
P=2/3ρc2
P=2/3ε(energy density)
Therefore (P=ωε) ω=2/3 (Nonrelativistic gas)
But for matter dominated universe we take ω=0
Why is it so??