# Hard Thermo Question: Find U, S, T For Part B & C

• mewmew
In summary, the conversation revolves around a problem from Reif's book, related to the concept of temperature in Schroeder's book. The participants have been able to solve part A of the problem, but are struggling with parts B and C. The concept of temperature is described as being related to the partial derivative of the number of ways to arrange atoms in a system, and the conversation also touches on the use of Sterling's approximation for factorials in solving the problem.
mewmew
Well, it's hard to me at least
http://img169.imageshack.us/img169/206/questionzg0.png
We are using Schroeders book and this problem was given to us from I assume the book by Reif. We haven't seem to have gotten far enough along in Schroeders book to have gone over this though.

I can do part A, but part B and C I am clueless on. I know that T = partial derivative of U/S but it doesn't seem to be helping me very much. Thanks in advance for any pointers.

Last edited by a moderator:
Ok, I found B more or less, by taking n=E/e. My equation for T looks pretty nasty because my multiplicity from part A was relatively nasty. I am still pretty much completely stuck on how to do C though.

What do you get for A? I get

$$\Omega(N,n)=\binom{N}{N-n}=\frac{N!}{n!(N-n)!}$$

(the number of ways to pick the position of the N-n atomes that are not in an interstititiisal position.

Temperature is $$\frac{1}{kT}=\frac{\partial \Omega(N,n)}{\partial E}$$.

I guess you said, $$n=n(E)=E/\epsilon$$, so then

$$\frac{\partial \Omega(N,n)}{\partial E}=\frac{\partial \Omega(N,n)}{\partial n}\frac{\partial n}{\partial E}= \frac{\partial \Omega(N,n)}{\partial n}\frac{1}{\epsilon}$$

But do you know how to take the derivative of a factorial?

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Sorry, I Should have been more specific. I took a Sterling approximation of the factorials of the form Ln(x!)= x Ln(x) - x, expanded each term, substituted in n = E/epsilon and differentiated. This was messy though and I am not sure the desired method.

## 1. What is the purpose of finding U, S, and T in a hard thermo question?

The purpose of finding U (internal energy), S (entropy), and T (temperature) in a hard thermo question is to determine the thermodynamic properties of a system. These properties are essential in understanding the behavior and changes of a system, and are often used in engineering and scientific calculations.

## 2. How do you solve for U, S, and T in a hard thermo question?

Solving for U, S, and T in a hard thermo question involves using the appropriate equations and principles of thermodynamics, such as the first and second laws of thermodynamics, to manipulate and solve for the unknown variables. It also requires a good understanding of the given system and its boundary conditions.

## 3. What information is needed to find U, S, and T in a hard thermo question?

To find U, S, and T in a hard thermo question, you need to have the values of other relevant thermodynamic properties, such as pressure, volume, and number of moles, as well as the boundary conditions of the system. Additionally, knowledge of the type of process (e.g. isothermal, adiabatic, etc.) may also be required.

## 4. Can U, S, and T be solved for in any type of system?

Yes, U, S, and T can be solved for in any type of system as long as the necessary information and equations are available. However, the specific method for solving for these variables may vary depending on the complexity and uniqueness of the system.

## 5. How do U, S, and T relate to each other in a system?

U, S, and T are all related to each other through the fundamental thermodynamic equations, such as the first and second laws of thermodynamics. For example, changes in U and S can affect each other, and changes in T can affect both U and S. These relationships are important in understanding the behavior of a system and predicting its changes.

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