Ch 20: The Micro/Macro Connection

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 20: The Micro/Macro Connection

Problem 71

Chapter 20, Problem 71

A thin partition divides a container of volume V into two parts. One side contains n_A moles of gas A in a fraction f_A of the container; that is, V_A = f_AV. The other side contains nB moles of a different gas B at the same temperature in a fraction f_B of the container. The partition is removed, allowing the gases to mix. Find an expression for the change of entropy. This is called the entropy of mixing.

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Start by recalling the formula for entropy change for an ideal gas when it expands or compresses: ΔS = nRln(Vf/Vi), where n is the number of moles, R is the gas constant, Vf is the final volume, and Vi is the initial volume.

For gas A, initially it occupies a volume VA = fA * V. After the partition is removed, it expands to occupy the entire container volume V. The change in entropy for gas A is ΔSA = nA * R * ln(V/VA). Substitute VA = fA * V into the equation to get ΔSA = nA * R * ln(1/fA).

Similarly, for gas B, initially it occupies a volume VB = fB * V. After the partition is removed, it also expands to occupy the entire container volume V. The change in entropy for gas B is ΔSB = nB * R * ln(V/VB). Substitute VB = fB * V into the equation to get ΔSB = nB * R * ln(1/fB).

The total change in entropy (ΔS_total) is the sum of the entropy changes for both gases: ΔS_total = ΔSA + ΔSB. Substitute the expressions for ΔSA and ΔSB to get ΔS_total = nA * R * ln(1/fA) + nB * R * ln(1/fB).

Simplify the expression for the total entropy change: ΔS_total = R * [nA * ln(1/fA) + nB * ln(1/fB)]. This is the final expression for the entropy of mixing.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Entropy

Entropy is a measure of the disorder or randomness in a system. In thermodynamics, it quantifies the amount of energy in a physical system that is not available to do work. The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, which implies that natural processes tend to move towards a state of maximum disorder.

Gibbs Free Energy

Gibbs Free Energy (G) is a thermodynamic potential that measures the maximum reversible work obtainable from a thermodynamic system at constant temperature and pressure. It is defined as G = H - TS, where H is enthalpy, T is temperature, and S is entropy. Changes in Gibbs Free Energy can predict the direction of chemical reactions and phase changes, with negative changes indicating spontaneous processes.

Mixing of Gases

The mixing of gases refers to the process where two or more different gases combine to form a homogeneous mixture. This process increases the entropy of the system because the number of accessible microstates increases as the gases intermingle. The entropy of mixing can be quantitatively expressed using the formula ΔS_mixing = -R(nA ln(fA) + nB ln(fB)), where R is the gas constant, and nA and nB are the number of moles of gases A and B, respectively.

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Related Practice

Textbook Question

Consider a container like that shown in Figure, with $n sub 1$ moles of a monatomic gas on one side and $n sub 2$ moles of a diatomic gas on the other. The monatomic gas has initial temperature $T sub 1 i$ . The diatomic gas has initial temperature $T sub 2 i$ . Show that the equilibrium thermal energies are

$\begin{aligned}\)E_{1f} &= $\frac{3n_1}{3n_1 + 5n_2}$ (E_{1i} + E_{2i}) $\E$_{2f} &= $\frac{5n_2}{3n_1 + 5n_2}$ (E_{1i} + E_{2i})\(\end{aligned}$

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Textbook Question

Consider a container like that shown in the Figure, with $n_1$ moles of a monatomic gas on one side and $n_2$ moles of a diatomic gas on the other. The monatomic gas has initial temperature $T_{1i}$ . The diatomic gas has initial temperature $T_{2i}$ . Show that the equilibrium temperature is

$T_{f} = \frac{3 n_{1} T_{1 i} + 5 n_{2} T_{2 i}}{3 n_{1} + 5 n_{2}}$

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Textbook Question

The rms speed of the molecules in 1.0 g of hydrogen gas is 1800 m/s. 500 J of work are done to compress the gas while, in the same process, 1200 J of heat energy are transferred from the gas to the environment. Afterward, what is the rms speed of the molecules?

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Textbook Question

The 2010 Nobel Prize in Physics was awarded for the discovery of graphene, a two-dimensional form of carbon in which the atoms form a two-dimensional crystal-lattice sheet only one atom thick. Predict the molar specific heat of graphene. Give your answer as a multiple of R.

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Textbook Question

An experiment you're designing needs a gas with γ = 1.50. You recall from your physics class that no individual gas has this value, but it occurs to you that you could produce a gas with γ = 1.50 by mixing together a monatomic gas and a diatomic gas. What fraction of the molecules need to be monatomic?

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Textbook Question

n₁ moles of a monatomic gas and n₂ moles of a diatomic gas are mixed together in a container. Derive an expression for the molar specific heat at constant volume of the mixture.

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