9. Langmuir isotherms: an ideal gas of particles is in contact with the surface of a catalyst (a) Show that the chemical potential of the gas particles is related to their temperature and pressure via μ = k&T [În (P/T³/²) + A₁], where A, is a constant. (b) If there are N distinct adsorption sites on the surface, and each adsorbed particle gains an energy & upon adsorption, calculate the grand partition function for the two-dimensional gas with a chemical potential μ. (c) In equilibrium, the gas and surface particles are at the same temperature and chemica potential. Show that the fraction of occupied surface sites is then given by f(T, P) = P/(P+Po(T)). Find Po(T).
9. Langmuir isotherms: an ideal gas of particles is in contact with the surface of a catalyst (a) Show that the chemical potential of the gas particles is related to their temperature and pressure via μ = k&T [În (P/T³/²) + A₁], where A, is a constant. (b) If there are N distinct adsorption sites on the surface, and each adsorbed particle gains an energy & upon adsorption, calculate the grand partition function for the two-dimensional gas with a chemical potential μ. (c) In equilibrium, the gas and surface particles are at the same temperature and chemica potential. Show that the fraction of occupied surface sites is then given by f(T, P) = P/(P+Po(T)). Find Po(T).
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Transcribed Image Text:(e) Using the characteristic function, show that
a(N)
(N²). = k„T
(f) Show that fluctuations in the number of adsorbed particles satisfy
(N*),
(N)?
1-f
Nf
![9. Langmuir isotherms: an ideal gas of particles is in contact with the surface of a catalyst.
(a) Show that the chemical potential of the gas particles is related to their temperature
and pressure via µ=k,T[In (P/T*2) + A], where A, is a constant.
(b) If there are N distinct adsorption sites on the surface, and each adsorbed particle
gains an energy e upon adsorption, calculate the grand partition function for the
two-dimensional gas with a chemical potential u.
(c) In equilibrium, the gas and surface particles are at the same temperature and chemical
potential. Show that the fraction of occupied surface sites is then given by f(T, P) =
P/(P+Po(T)). Find Po(T).
(d) In the grand canonical ensemble, the particle number N is a random variable. Cal-
culate its characteristic function (exp(-ikN)) in terms of Q(Bu), and hence show
that
(N"), =-(kgT)"-1
where G is the grand potential.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf8a0b14-ded2-434a-960f-d94e1fea6f06%2F888653d0-3deb-44a9-8c98-4ae70d3bf269%2Fwgqrsz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:9. Langmuir isotherms: an ideal gas of particles is in contact with the surface of a catalyst.
(a) Show that the chemical potential of the gas particles is related to their temperature
and pressure via µ=k,T[In (P/T*2) + A], where A, is a constant.
(b) If there are N distinct adsorption sites on the surface, and each adsorbed particle
gains an energy e upon adsorption, calculate the grand partition function for the
two-dimensional gas with a chemical potential u.
(c) In equilibrium, the gas and surface particles are at the same temperature and chemical
potential. Show that the fraction of occupied surface sites is then given by f(T, P) =
P/(P+Po(T)). Find Po(T).
(d) In the grand canonical ensemble, the particle number N is a random variable. Cal-
culate its characteristic function (exp(-ikN)) in terms of Q(Bu), and hence show
that
(N"), =-(kgT)"-1
where G is the grand potential.
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