Consider N atoms con¯ned on a surface of area A at temperature T . The atoms
form a two-dimensional (2D) gas of classical, noninteracting particles.
(a) Calculate the partition function for the system.
(b) Calculate the Helmholtz free energy, F , of the gas. Compare it with the 3D
case.
(c) Calculate the internal energy of the gas. Compare it with the 3D case.
(d) Calculate the surface tension of the gas, ° = (@F=@A)T;N .
(e) Calculate the momentum distribution n(p) which determines the number of
atoms N(p) with momenta between p and p + dp: N(p) = n(p)dp
In a volume d^2p of momentum space, there are A d^2p/h^2 states. The partition function for one atom can thus be written as:
Z1 =
A/h^2 Integral d^2p exp[-beta p^2/(2m)]
Integrate over the angle, leaving the intergral over the magnitude of the momentum:
Z1 =
2 pi A/h^2 Integral from 0 to infinity of dp p exp[-beta p^2/(2m)] =
2 pi A/h^2 m/beta = A 2 pi m k T/h^2
The partition function for N atoms is thus given by:
Z = Z1^N/N! =
(A 2 pi m k T/h^2)^N/N!
The free energy is minus k T Log(Z) and the pressure and surface tension can be computed from this by carying out the differentiations, which are trivial.
To answer these questions, we need to apply the principles of statistical mechanics for a classical, non-interacting 2D gas.
(a) Partition Function:
The partition function, denoted by Z, determines the thermodynamic properties of a system. For a 2D system, we can write the partition function as:
Z = 1/N! * (1/h^2) * ∫e^(-(p^2/2m) / (kT)) dp^(2N)
Here, N is the number of atoms, h is the Planck's constant, p is the momentum, m is the mass of each atom, k is the Boltzmann constant, and T is the temperature.
To calculate the partition function, you need to perform the integration over all possible momenta for all N atoms. The limits of integration need to be determined based on the specific conditions of the problem (e.g., size of the system and confinement).
(b) Helmholtz Free Energy:
The Helmholtz free energy, F, is related to the partition function by the equation:
F = -kT * ln(Z)
So once you calculate the partition function, you can use this equation to find the Helmholtz free energy of the 2D gas. To compare with the 3D case, you would need to perform similar calculations for a 3D gas.
(c) Internal Energy:
The internal energy, U, can be derived from the Helmholtz free energy:
U = F + TS
Here, S is the entropy of the system, and T is the temperature. If you know the entropy, you can use this equation to calculate the internal energy in terms of the Helmholtz free energy.
(d) Surface Tension:
The surface tension, σ, can be calculated using the derivative of the Helmholtz free energy with respect to the surface area, A:
σ = (∂F/∂A)T,N
To calculate this derivative, you would need to differentiate the expression for the Helmholtz free energy with respect to A while keeping T and N constant.
(e) Momentum Distribution:
The momentum distribution, n(p), is determined by the system's velocity distribution function. To calculate it, you would need to integrate the probability distribution function over the specified momentum range. The number of atoms, N(p), is then given by the product of this distribution function and the momentum differential, dp.
To summarize, to answer these questions, you would need to perform calculations involving integrals, derivatives, and logarithms based on the provided equations. The specific values and limits of integration depend on the given conditions of the problem.