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### Molar Heat Capacity of a Gas

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Date: 11/17/1999 at 01:23:06
From: Wiley Ang
Subject: Thermodynamics/Physics

Dear Dr. Math

I recently took part in the National Physics Competition in my country
but I got stuck on the following question:

The volume of n moles of a monatomic perfect gas with an initial
volume V and an initial temperature T is halved at a constant rate.
At the same time, its temperature is doubled at a constant rate.

(a) Prove that the work done to the gas is given by

W = nRT*(3*log[2]-1)

(b) Obtain an expression for the heat absorbed during the process

(c) Prove that the molar heat capacity of this gas during the process
is 3.49 J/mol/K.

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```
Date: 11/17/1999 at 06:35:07
From: Doctor Mitteldorf
Subject: Re: Thermodynamics/Physics

Dear Wiley,

Since the question defines T and V as the (constant) initial
temperature and volume, let's use t and v for the variable quantities
as the process occurs. The phrase "constant rate" assures us that
change in temperature is proportional to change in volume. Since the
overall change in volume is V/2 (from V down to V/2) and the overall
change in temperature is T (from T up to 2T), we can say that

delta t = (2T/V)*delta v.

We can write this as

(t-T)=(2T/V)*(V-v)

So the temperature follows a straight line defined by

t = T+2T(1-v/V)=T(3-2v/V)

We're interested in the pressure as the gas is compressed, because it
is the pressure that determines the work. P = nRt/v. Let's substitute
our expression for t in terms of v:

P = nRt/v

= nRT(3-2v/V)/v

= 3nRT/v - 2nRT/V

Work is the integral of pressure with respect to volume, so

V
W = Int[3nRT/v - 2nRT/V]
V/2

If you solve this integral, you should get the result in (a). For
part (b), you want to know the heat absorbed by the gas during the
entire process. First find the energy at the end minus the energy at
the beginning: that is 3/2 nR(2T) at the end and 3/2 nRT at the
beginning. Then subtract the work that was done, calculated in (a).
The difference is the heat that was added. Converting the answer from
(b) to a number should give you the answer for (c).

Will you write back and let me know if it works?

- Doctor Mitteldorf, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Physics/Chemistry

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