GREPhysics.NET: GR8677 #14

GR8677 #14

Problem

GREPhysics.NET Official Solution

Thermodynamics $\Rightarrow$ }Exact differentials

The key equation is: $PV=nRT$ , and its players, $P,V,n,R,T$ , are terms one should be able to guess.

(A) True, according to the ideal gas law. (This is also the final step in deriving Mayer's Equation, as shown below.)

(B) This translates into the statement $\left.\frac{dq}{dT}\right|_V=\left.\frac{dq}{dT}\right|_P\Rightarrow c_V=c_P$ .The problem gives away the fact that for an ideal gas $C_P \neq C_V$ . B can't be right.

(C) According to the ideal gas law, the volume might change.

(D) False. An ideal gas's internal energy is dependent only on temperature. More elegantly, $u=u(T)$ .

(E) Heat needed for what?

If one is interested in the formal proof of the relation $c_p=c_v+nR$ , read on about Mayer's equation:

For thermo, in general, there's an old slacker's pride line that goes like, ``When in doubt, write a bunch of equations of states and mindlessly begin taking exact differentials. Without exerting much brainpower, one will quickly arrive at a brilliant result." Doing this,

$\begin{eqnarray} Q=U+W&\Rightarrow& dQ=dU+PdV\\ U=U(T,V)&\Rightarrow& dU=\left.\partial_T U\right|_T +\left.\partial_V U\right|_V\\ PV=nRT&\Rightarrow&PdV+VdP=nRdT<br /> \end{eqnarray}$

Plugging in the first law of thermodynamics into the $U$ equation of state, one gets $dQ=\left.\partial_T U\right|_T+\left(\left.\partial_T U\right|_V+P\right)dV=\left.\partial_T U\right|_T+PdV$ , where the last simplification is made by remembering the fact that the internal energy of an ideal gas depends only on temperature.

(Taking the derivative with respect to T at constant volume, one gets $\left.\frac{dQ}{dT}\right|_V=\left.\partial_T U\right|_T=C_V$ .)

Plugging in the simplified result for $dQ=...$ into the third equation of state, the ideal gas equation, one gets: $dQ-C_vdT+VdP=nRdT$ . Taking the derivative at constant pressure, one gets:

$\left.\frac{dQ}{dT}\right|_P=C_V+nR<br />$

So, one sees that it is the ideal gas equation that makes the final difference. The work of an ideal gas changes when temperature is varied.

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Comments

kevglynn
2006-10-31 09:53:04

I love Stat Mech, and one of the reasons is probably because I love the book that my professor used/wrote for the course. It's awesome, and everyone should check it out:

Elementary Lectures in Statistical Mechanics

by George D. J. Phillies

ISBN: 0387989188

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matno
2005-10-31 20:58:48

What is a good book on Stat Mech???

yosun
2005-11-01 02:30:26

The book that I started out with was Classical and Statistical Thermodynamics by A. Carter. On the Stat part, it has a decent introduction to the distributions and the combinatorics involved in deriving them. Further understanding of probability can be gained from Reif. Kittel, Kittel and Kroemer and Ashcroft are also good supplements (three seperate books).

yosun
2005-11-01 16:06:25

More on Stat Mech textbooks: Sturge offers a good conceptual intro for Bose-Einstein Condensation.

Richard
2007-11-02 11:11:00

I won't say it's GREAT, but it made for a fun read when I took the course:
An Introduction to Thermal Physics, by Daniel V. Schroeder.

The problems are hellish though.

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