Boltzmann statistics - finding the number of particles

[jianxu]

Boltzmann statistics -- finding the number of particles

Homework Statement

A column of water contains fine spherical metal particles of radius 2 x10^-8m, which are in thermal equilibrium at 25C. If there are 1000 particles per unit volume at a given height, how many particles would be found in the same volum 1 mm higher? The density of the metal is 2 x10⁴ kg m^-3. Hint: use gravitational force and buoyancy in water and compute change in potential energy

Homework Equations

F_net= $$\rho$$Vg - mg ($$\rho$$ is density, V is volume, g is gravity, m is mass)

F*d = -mg $$\Delta$$h (h is height, d = height, F = F_net

n(E) = g(E)f_b(E) = A g(E)e^-E/kT
n(E) = number of particles with energy E
g(E) = statistical weight of the state with energy E
A = normalization constant whose value depends of the system
k = Boltzmann constant
T = temperature
E = energy

The Attempt at a Solution

What I did was just substitute F_net= $$\rho$$Vg - mg
into F*d = -mg $$\Delta$$h which gives me the change in potential energy. I'm not sure what to do with the boltmann distribution formula after that or do I even need it? I'm just stuck and don't know where to go. Any help will be appreciated

Thank you

 

[olgranpappy]

Homework Statement

A column of water contains fine spherical metal particles of radius 2 x10^-8m, which are in thermal equilibrium at 25C. If there are 1000 particles per unit volume at a given height, how many particles would be found in the same volum 1 mm higher? The density of the metal is 2 x10⁴ kg m^-3. Hint: use gravitational force and buoyancy in water and compute change in potential energy

Homework Equations

F_net= $$\rho$$Vg - mg ($$\rho$$ is density, V is volume, g is gravity, m is mass)

F*d = -mg $$\Delta$$h (h is height, d = height, F = F_net

n(E) = g(E)f_b(E) = A g(E)e^-E/kT
n(E) = number of particles with energy E
g(E) = statistical weight of the state with energy E
A = normalization constant whose value depends of the system
k = Boltzmann constant
T = temperature
E = energy

The Attempt at a Solution

What I did was just substitute F_net= $$\rho$$Vg - mg
into F*d = -mg $$\Delta$$h which gives me the change in potential energy. I'm not sure what to do with the boltmann distribution formula after that or do I even need it? I'm just stuck and don't know where to go. Any help will be appreciated

Thank you

you need the density as a function of height. that's why you need the boltmann distribution function for the density.