## shubham 3 years ago The Principles of Statistical Mechanics :- The exponential atmosphere. If the temp. is same at all heights, the problem is to discover by what law the atmosphere becomes tenuous as we go up. If N is no. of molecules in a volume V of gas at pressure P, then we know PV = NkT, or P = nkT, where n =N/V is the no. of molecules per unit volume, if we know the no. of molecules per unit volume, we know pressure and vice-versa : they are proportional to each other, since the temperature is constant. But the pressure is not constant, it must increase as the altitude is reduced. If we take a unit a

1. shubham

at height h, then the vertical force from below is pressure P. The vertical force pushing down at height h+dh would be same, in absence of gravity. so, dn/dh = -mgn/kt (tells how density goes down as we go up in energy) => n = n0. e^ ( -mgh/kt) I want you to explain this to me.

2. ramkrishna

if you are reading from Feynmen lectures then you know dP=-mgndh also we have P=nkT so, dP=kTdn using both equation we have kTdn=-mgndh or dn/n=-mgdh/(kT) on integrating we get ln(n)=-mgh/(kT) + constant or n=exp(-mgn/(kT))* exp(constant) here exp(constant)=n0

3. shubham

@ramkrishna Yes sir, this is an excerpt from Feynmen lectures. I was looking for the physical significance of Boltzman factor, e ^ {-( E2 - E1 ) / kT }. Can you please explain that ..

4. ramkrishna

For a system in equilibrium at temperature T, the probability that the system is in a particular quantum state i, a particular microstate, with energy \[E_i\] is proportional to e^{-Ei/kT} The ratio of probabilities of the particles in the state 1 and 2 is n1/n2=e ^ {-( E2 - E1 ) / kT }