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
(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

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  1. 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.

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