Physics

Physics · thermodynamics

Problem

f. What is the increase in the entropy of the universe as a result of the free expansion? Give your answer to first order in

\frac{a^{\prime} N}{V k_{B} T}

. Given constants: ## Real Expansion Consider a "real" monatomic gas consisting of

N

atoms of negligible volume and mass

m

in equilibrium inside a closed cubical container of volume

V

. In this "real" gas, the attractive forces between atoms is small but not negligible. Because these atoms have negligible volume, you can assume that the atoms do not collide with each other for the entirety of the problem. Context question: a. Consider an atom in the interior of this container of volume

V

. Suppose the potential energy of the interaction is given by

u(r)= \begin{cases}0 & r<d \\ -\epsilon\left(\frac{d}{r}\right)^{6} & r \geq d\end{cases}

where

d \ll V^{1 / 3}

is the minimum allowed distance between two atoms. Assume the gas is uniformly distributed within the container, what is the average potential energy of this atom? Write your answer in terms of

a^{\prime}=\frac{2 \pi d^{3} \epsilon}{3}, N

, and

V

. Context answer:

-2 a^{\prime} N / V

Context question: b. What is the average potential energy of an atom near the boundary of the box? Assume that there is no interaction between atoms near the boundary and the box itself. Context answer:

-a^{\prime} N / V

Context question: c. Using Bernoulli's law

P+U+\rho v^{2} / 2=

constant, with pressure

P

, potential energy density

U

, mass density

\rho

and fluid velocity

v

, what is the pressure at the boundary of the box? Assume the interior pressure is given by the ideal gas law. Context answer:

\frac{N k T}{V}-a^{\prime} \frac{N^{2}}{V^{2}}

Context question: d. Assuming most atoms are in the interior of the box, what is the total energy of the atoms in the box? Context answer:

\frac{3}{2} N k T-a^{\prime} N^{2} / V

Extra Supplementary Reading Materials: Now consider an insulated partitioned container with two sections, each of volume

V

. We fill one side of the container with

N

atoms of this "real" gas at temperature

T

, which the other side being a vacuum. We then quickly remove the partition and let the gas expand to fill the entirety of the partitioned container. During this expansion, the energy of the gas remains unchanged. Context question: e. What is the final temperature of the gas after the expansion? Context answer:

T^{\prime}=T-\frac{a^{\prime} N}{3 k_{B} V}

Related theory

thermodynamics

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