Question

Consider an ideal monatomic gas og N indistinguishable particles, mass m each and total energy U, confined to a one dimnensional channel of length L.

Calculate the multiplicity of the 1-D gas and its entropy, following the line of reasoning used in deriving these quantites in the 3-D case.

Answer 1

Entropy S of a Monoatomic ideal gas equation called the Sackur-Tetrode equation.

N = number of atoms

k = Boltzmann's constant = 1.3806 X 10^-23 J/K

V = volume

U = internal energy

h = Planck's constant = 6.626 X 10^-34 J/sec

One of the things which can be determined directly from this equation is the change in entropy during an isothermal expansion where N and U are constant (implying Q=W). Expanding the entropy expression for V_f and V_i with log combination rules leads to

For determining other functions, it is useful to expand the entropy expression to separate the U and V dependence.

Then making use of the definition of temperature in terms of entropy:

This gives an expression for internal energy that is consistent with equipartition of energy.

with kT/2 of energy for each degree of freedom for each atom.

For processes with an ideal gas, the change in entropy can be calculated from the relationship

Making use of the first law of thermodynamics and the nature of system work, this can be written

This is a useful calculation form if the temperatures and volumes are known, but if you are working on a PV diagram it is preferable to have it expressed in those terms. Using the ideal gas law

Then

But since specific heats are related by C_P = C_V + R,

Energy

If we denote the average energy áEñ then

We use the notation that áEñ = U - U(0) where U(0) is the energy at zero Kelvin. Recalling that b = 1/kT this can be rewritten as

This can be written compactly as

Heat Capacity

The heat capacity is a coefficient that gives the amount of energy to raise the temperature of a substance by one degree Celsius. The heat capacity can also be described as the temperature derivative of the average energy. The constant volume heat capacity is defined by

using the notation that áEñ = U - U(0) where U(0) is the energy at zero Kelvin. The molar internal energy of a monatomic ideal gas is áEñ = 3/2RT. The heat capacity of a monatomic ideal gas is therefore C_v = 3/2R.

For a monatomic gas there are three degrees of freedom per atom (these are the translations along the x, y, and z direction). Each of these translations corresponds to ½RT of energy. For an ideal triatomic gas some of the energy used to heat the gas may also go into rotational and vibrational degrees of freedom. For solids there is no translation or rotation and therefore the entire contribution to the heat capacity comes from vibrations. Given their extended nature the vibrations in solids are much lower in frequency than those of gases. Therefore, while vibrations in typical triatomic gases typically contribute little to the heat capacity, the vibrational contribution to the heat capacity of solids is the largest contribution. As the temperature is increased, there are more levels of the solid accessible by thermal energy and therefore Q increases. This also means that U increases and finally that C_v increases. In the high temperature limit in an ideal solid there are 2N vibrational modes that are accessible giving rise to a contribution to the molar heat capacity of 2R.

Pressure

Pressure can also be derived from the canonical partition function.

The average pressure is the sum of the probability times the pressure

From thermodynamics, pressure is expressed as

so we can write

In a few steps we can show that the temperature can be expressed in terms of the partition function.

The derivative of the partition function with respect to volume is

The average pressure can then be written as

Which shows that the pressure can be expressed solely terms of the partition function.

We can use this result to derive the ideal gas law. For N particles of an ideal gas

where

is the translational partition function. The utility of expressing the pressure as a logarithm is clear from the fact that we can write

We have used the property of logarithms that ln(AB) = ln(A) + ln(B) and ln(X^Y) = Yln(X). Only one term in the ln Q depends on V.

Taking the derivative of NlnV with respect to V gives

Substituting this into the above equation for the pressure gives P=NkT/V which is the ideal gas law. Recall that Nk = nR where N is the number of molecules and n is the number of moles. R is the universal gas constant (8.314 J/mol-K) which is nothing more than k multiplied by Avagadro’s number. N_Ak = R converts the constant from a "per molecule" to a "per mole" basis.

Entropy

We have calculated E = U – U(0), which is the internal energy referenced to the value, U(0) at absolute zero (T = 0 K).

We can now calculate the entropy, S = k ln W

Now recalling the definition of the Boltzmann distribution

ln p_i = - be_i – ln q

The entropy is,

.

The entropy can be expressed in terms of the system partition function Q

.

Helmholtz Free Energy

The Helmholtz free energy is A = U - TS. Substituting in for U and S from above we have

A = - kT ln Q.