Callen: Thermodynamics (general fail)

[go_ducks]

Greetings, I'm a 4th year physics undergrad, hoping to avoid entering a 5th. Using 2nd edition of Callen for Thermo class in upper div. I'm having a real hard time solving any problem in this book. I haven't had nightmares over homeworks since my math methods class which used Arfken. The problem is just ..I don't seem to have the pre-requisite understanding of thermodynamics / statistical physics. Lecture also covers different topics (yes, really.) I am going to post up a couple problems, not really looking for answers but maybe what I should've read up on first before I tried this problem. This is from the 2nd edition of callen. I have a lot of other problems I just think i'd ask about the worst one first.

Homework Statement

19.3-6. Consider a small quantity of matter consisting of a fixed number N moles in a large fluid system. Let ρN be the average density of these N moles: the mass divided by the volume. Show that equation 19.17 implies that the density fluctuations are

<(ΔρN)^2> / (ρN)^2 = + kb * T * κt / V

in which V is the average volume of the N moles.

--
ok , i can't seem to understand the sigma symbol, so let me explain

kb is the Boltzmann constant
κt is k-sub-t (subscript), i actually don't know what this is.
ΔρN has a hat ( a ^ symbol) on the ρ
ρN is ρ-sub-N (subscript)

Homework Equations

eq 19.17 is:

<(ΔV-hat)^2> = -kb T (dv/dp)t,N1,... = kB T V κT

The Attempt at a Solution

I met up with 4 other students and we tried to make sense of it. It doesn't . Read the Callen chapter which is bewildering. Other than ch 19, we went over ch 1-7. I tried reading the other chapters before it. Went home trying to work on it. Tried to just replace ρ with M/V throughout the problem, hoping it would be a plug-and-chug answer. Tried to reverse engineer the goal-equation to see what needs doing (it seems like multiply both sides by M/V^3... can't recall off-hand.. it was something that made no sense. After that I began reading about fluctuations and stat mech online... also didnt help. Basically after that I started trying to just look the answer up somewhere, and that wasnt fruitful. I have..an empty sheet, because I don't even know where to start. I considered maybe attacking the (dv/dp) term in 19.17 since we have been doing manipulations like that up until now, but I am not very sure of what I am doing with that anyway.So..I'm sure someone knows how to nail this problem. Can I be provided with some maybe useful links and advise on how to solve it? I mean, I feel like I just don't have the tools. Maybe there is a concept of thermodynamics I missed badly along the way somewhere. I love my professor but wow this problem has made me suffer.

 

[vela]

Equation 19.17 is
$$\langle (\Delta \hat{V})^2 \rangle = -k_B T\left(\frac{\partial V}{\partial P}\right)_{T, N_1, \dots} = k_B T V \kappa_T$$where
$$\kappa_T = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T$$is the isothermal compressibility (see page 84).

The density is equal to $\hat{\rho}_N = N/\hat{V}$, and its average value is $\rho_N = N/V$, where V is the average volume. Therefore, the fluctuation in density is
$$\Delta \hat{\rho}_N = \hat{\rho}_N - \rho_N = \frac{N}{\hat{V}} - \frac{N}{V}$$Try using $\hat V = V + \Delta \hat V$ and expanding the first term to first order in $\Delta \hat V$.

 

[go_ducks]

An expansion! I should've known...