Consider a mono-atomic ideal gas, under the action of a gravitational field, located at a height...

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Consider a mono-atomic ideal gas, under the action of a gravitational field, located at a height y above the surface of the earth, and inside a box with volume V = L2Ay. That is, the single-particle Hamiltonian is given by H({p,q}) = +mgy, 2m (2.1) where p is the particle's momentum, m their mass, and g is the gravity. (1) Assuming that the gas is at constant temperature T, show that the canonical single-particle partition function is given by L Z = e (2.2) (2) Find the expression for the grand canonical partition function as a function of T, V, and the fugacity z = e (3) Find the average number of particles (N) as a function of z, T, V, and the chemical potential as a function of (N), T, V. (4) The previous expressions are a rough approximation of the Earth's atmosphere at a specific layer. Now consider two consecutive layers of the atmosphere at heights y and y + Ay, with average number of particles (N(y)) and (N(y + Ay)) respectively. Assuming that the two layers are in thermal and chemical equilibrium, in the limit of Ay >> 0, find how the average number of particles varies as a function of the height. Consider a mono-atomic ideal gas, under the action of a gravitational field, located at a height y above the surface of the earth, and inside a box with volume V = L2Ay. That is, the single-particle Hamiltonian is given by H({p,q}) = +mgy, 2m (2.1) where p is the particle's momentum, m their mass, and g is the gravity. (1) Assuming that the gas is at constant temperature T, show that the canonical single-particle partition function is given by L Z = e (2.2) (2) Find the expression for the grand canonical partition function as a function of T, V, and the fugacity z = e (3) Find the average number of particles (N) as a function of z, T, V, and the chemical potential as a function of (N), T, V. (4) The previous expressions are a rough approximation of the Earth's atmosphere at a specific layer. Now consider two consecutive layers of the atmosphere at heights y and y + Ay, with average number of particles (N(y)) and (N(y + Ay)) respectively. Assuming that the two layers are in thermal and chemical equilibrium, in the limit of Ay >> 0, find how the average number of particles varies as a function of the height.