Latent heat, liquid-gas transition, stat mech with gravity

[mchan1014]

Problem solved.

Homework Statement

The boiling point of a certain liquid is 95^OC at the top of a mountain
and 105^OC at the bottom. Its latent heat is 1000 cal/mole. Calculate the
height of mountain.
It's Q4 of chapter 15 in Statistical Mechanics by S.K. Ma

Homework Equations

it should be the Clausius-Clapeyron Equation

The Attempt at a Solution

I tried to use the canonical partition function to derive the pressure in terms of height, and then plug it into the Clausius-Clapeyron Equation, but it is not successful.

 

[mchan1014]

the book by S.K. Ma here : http://books.google.com/books?id=3X...#v=onepage&q=inauthor:"Shang-keng Ma"&f=false

 

[mchan1014]

http://books.google.com/books?id=3X...TW&source=gbs_toc_r&cad=4#v=onepage&q&f=false

 

[mchan1014]

as the pressure at height z is P(z) = P(0) exp ( - mgz / kT )
I plug it into the Clausius-Clapeyron Equation and found that it depends on m, the mass of the liquid molecule, which is not given

 

[paranormal]

Can anyone help me solve this problem?
This is a very interesting problem that combines concepts from statistical mechanics, thermodynamics, and gravity. To solve this problem, we first need to understand the relationship between temperature, pressure, and height in a gravitational field.

In a gravitational field, the pressure at a certain height is given by the hydrostatic equation: P = P0 + ρgh, where P0 is the pressure at sea level, ρ is the density of the liquid, g is the acceleration due to gravity, and h is the height. This equation assumes that the density of the liquid remains constant, which is a reasonable assumption for most liquids.

Next, we can use the canonical partition function to calculate the pressure in terms of temperature and height. The partition function for a system with gravity can be written as Q = Q0exp(-mgz/kT), where Q0 is the partition function without gravity, m is the mass of a molecule, z is the height, k is the Boltzmann constant, and T is the temperature.

Combining the hydrostatic equation and the partition function, we can write the pressure at a certain height as P = P0 + ρgh = P0 + ρg/kT * ln(Q/Q0). Now, we can plug this expression into the Clausius-Clapeyron Equation, which relates the boiling point of a liquid to its latent heat and pressure.

Using the given information, we can set the boiling point at the top of the mountain (95OC) equal to the boiling point at the bottom (105OC) and solve for the height h. This will give us the height of the mountain.

I hope this helps you solve the problem. Good luck!