thnxs!!!!

Expected value of W:

Variance of W:

If q<1 then

In the problem and therefore we require that ; thus when we have closed sum

On the other hand,

Because

To find required we have

or

Equation (10) has only one solution(this can be shown comparing derivatives of rhs and lhs functions):

To satisfy requirement of we give result in limit form:

QED

1st law:

Enthalpy definition:

Enthalpy differential:

Maxwell equation from (2):

Definition of heat capacity and insertion (1) into definition:

Write entropy as function of T and v:

Derive expression for entropy differential and use (4) and (5):

Insert (7) into (1):

QED

Part 2

van der Waals equation:

Express p:

At constant T equation (8) becomes:

Differentiate (10) at constant v and put into (11):

QED

Derive the principal thermodynamic connection formulas for this ensemble.

When i try to derive the partition function I am getting a term that makes no sense. I was wondering if anyone could help me?

thanks

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All the best,
Dr. Carl McBride.

P/KT = rho(1) + rho(2) + B20(T)*rho(1)^2 + B11(T)* rho(1)*rho(2) + B02*rho(2)^2 +…………….
rho =density(=N/V)

and find an expression for the coefficients B20(T), B11(T), B02(T).

again show that if the virial expansion is written in the form

P/KT = rho + B2(T)* rho^2+ …………………..

with rho= rho(1) + rho(2) , then

B2(x,T)= (x1^2)*B20(T) + x1*(1-x1) * B11(T) + ((1-x1)^2)* B02(T)

where x1= mole fraction of component 1.

From the thermodynamic potential:

for constant N (number of molecules) we can write:

And from Maxwell Transformation:

By substituting equation (4) into equation (1) we get:

Which is true as:

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Julian Talbot

The most probable value of the molecular speed occurs when the derivative of the function is equal to 0

Solving for v;

The mean v is determined, as shown below, where is utilized

The root-mean-square speed is computed as follows, where is utilized

In the isothermal-isobaric ensemble, the partition function can be written as

From which the average enthalpy can be obtained as

Therefore

where we have used the definitions and

Relations between the two sets of virial coefficients can be determined as follows;

First solve equation 1 for p and substitute into equation 2 as follows;

Collecting terms within equation 3 with common powers of V,

In order to write equation 4 as equation 1, the following must be true,

The low temperature limiting form of the heat capacity according to the Einstein model is as follows;

Make the proper substitutions provided within the question in order to convert Cartesian coordinates to a spherical polar coordinate representation;

Integrate equation 2 over all space with respect to and

For the fraction of molecules with speeds between v and v + dv, substitute p=mv to get

is the same as for a three-dimensional diatomic gas;

with a degeneracy of 2 for all J except J=0, for which the degeneracy is 1;

Since is the same as a three dimensional diatomic gas and the is q(T) becomes

The average energy becomes

Start with:

Using the relationship;

Also,

Rearranging equation 6 and substituting into equation 5 results in

Rearranging equation 7 results in

Substitute equation 8 into equation 1 to get

which is always true

Show for bosons

Using the same methodology as above the following equations are derived

Substituting equation 12 into equation 11 results in

Substituting equation 13 into equation 10 results in

Rearranging equation 14 results in

Which is always true

Substitute giving

This suggests the gamma function. So let or
Substituting gives

Using the definition of the gamma function finally gives

If then

Utilizing Euler's Theorem;

Take the derivative of both sides with respect to ;

Set equal to 1;

Rearrange the equation;

Substitute into equation 1 to get

The terminal velocity is attained when the net force on the body is zero. This gives

To solve the equation of motion we use the result that

Integrating gives

Integrating again gives

For large t, this gives