I need help starting it...I think if I understand how to start the problem I can solve it...and probably part b too.

 

The partition function, q, can be used to determine the thermodynamic properties of a system. During the lecture for chapter 12, we have derived, as an example, the partition function of a simple two-state system. We have also used q to determine the internal energy for this system.

An important aspect of thermodynamics is to know the internal energy for a system of particles that can translate (move) in free space, but that cannot vibrate or rotate (an ideal gas would be such a simple model system).

a) Please derive the partition function, q, for translation in one dimension. This can be done by summing up over all the Boltzmann exponents of the energy levels determined for the one-dimensional particle in the box system:

En = n2h2/(8mL2) (Eq. 9.7 in textbook)

Hint: Substitute the sum in q for an integral and use the following integral:

 

b) Based on the partition function of the particle in the box system from a), derive an equation for the internal energy of the system using the following equation:

(Eq. 12.19 in textbook)

Hint: To obtain the derivative dq/dT use d(T1/2)/dT = ½ T-1/2

How does your answer compare to the equation for the temperature dependence of the internal energy of an ideal gas stated in the lectures for chapter 1?

Don’t hesitate to ask me if you have problems with the solution.

 

Or not... haha

Well, you could get started by writing out the sum. Incidentally, you can put equations with all the fancy symbols if you use [ math] [ /math] tags (without the space after the [)

 

Oh, and I think your title is wrong. There seems to be no biological aspects here.