Consider a monatomic ideal gas that lives at a height z

Chapter 3, Problem 37P

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QUESTION:

Consider a monatomic ideal gas that lives at a height \(z\) above sea level, so each molecule has potential energy \(mgz\) in addition to its kinetic energy.

(a) Show that the chemical potential is the same as if the gas were at sea level, plus an additional term \(mgz\)

\(\mu(z)=-k T \ln \left[\frac{V}{N}\left(\frac{2 \pi m k T}{h^{2}}\right)^{3 / 2}\right]+m g z\)

(You can derive this result from either the definition \(\mu=-T(\partial S / \partial N)_{U, V}\) or the formula \(\mu=(\partial U / \partial N)_{S, V}\).)

(b) Suppose you have two chunks of helium gas, one at sea level and one at height \(z\), each having the same temperature and volume. Assuming that they are in diffusive equilibrium, show that the number of molecules in the higher chunk is

\(N(z)=N(0) e^{-m g z / k T}\)

in agreement with the result of Problem 1.16

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QUESTION:

Consider a monatomic ideal gas that lives at a height \(z\) above sea level, so each molecule has potential energy \(mgz\) in addition to its kinetic energy.

(a) Show that the chemical potential is the same as if the gas were at sea level, plus an additional term \(mgz\)

\(\mu(z)=-k T \ln \left[\frac{V}{N}\left(\frac{2 \pi m k T}{h^{2}}\right)^{3 / 2}\right]+m g z\)

(You can derive this result from either the definition \(\mu=-T(\partial S / \partial N)_{U, V}\) or the formula \(\mu=(\partial U / \partial N)_{S, V}\).)

(b) Suppose you have two chunks of helium gas, one at sea level and one at height \(z\), each having the same temperature and volume. Assuming that they are in diffusive equilibrium, show that the number of molecules in the higher chunk is

\(N(z)=N(0) e^{-m g z / k T}\)

in agreement with the result of Problem 1.16

ANSWER:

Step 1 of 8

In terms of entropy, the chemical potential is described as follows:

\(\mu=-T\left(\frac{\partial S}{\partial N}\right)_{U, V} \ldots \ldots \ldots \ldots(1)\)

This definition leads to a general thermodynamic identity:

\(d U=T d S-P d V+\mu d N\)

We can get another formula for  from this:

\(\mu=\left(\frac{\partial U}{\partial V}\right)_{S, V}\)

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