- #1
chingel
- 307
- 23
Why can't we use the Boltzmann distribution to calculate what fraction of particles are in the liquid and gas states?
If in liquid state the potential energy is smaller by amount ##L## per mole lets say, the using the Boltzmann distribution (##r\propto \exp(-\Delta E/RT)##) the fraction of particles in the liquid state would be $$r_l = \frac{\exp(L/RT)}{1+\exp(L/RT)}$$ and for gas $$r_g = \frac{1}{1+\exp(L/RT)}$$
However this is not true since I can actually have any amount of water and vapor in equilibrium at some given temperature, e.g. half the container full of water or only quarter full of water (the remaining is vapor), since as long as the pressure is the saturated vapor pressure it should be in equilibrium. Why doesn't the Boltzmann distribution apply in this case?
Because in a different example, to calculate how does the density of air change with height in an isothermal atmosphere, I can use the potential energy per mole ##\mu gh## and the Boltzmann distribution to get that the fraction of particles at height ##h## is proportional to ##\exp(-\mu gh/RT)##, which is correct. Why does it work here, but not in the previous case?
If in liquid state the potential energy is smaller by amount ##L## per mole lets say, the using the Boltzmann distribution (##r\propto \exp(-\Delta E/RT)##) the fraction of particles in the liquid state would be $$r_l = \frac{\exp(L/RT)}{1+\exp(L/RT)}$$ and for gas $$r_g = \frac{1}{1+\exp(L/RT)}$$
However this is not true since I can actually have any amount of water and vapor in equilibrium at some given temperature, e.g. half the container full of water or only quarter full of water (the remaining is vapor), since as long as the pressure is the saturated vapor pressure it should be in equilibrium. Why doesn't the Boltzmann distribution apply in this case?
Because in a different example, to calculate how does the density of air change with height in an isothermal atmosphere, I can use the potential energy per mole ##\mu gh## and the Boltzmann distribution to get that the fraction of particles at height ##h## is proportional to ##\exp(-\mu gh/RT)##, which is correct. Why does it work here, but not in the previous case?