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 72

Chapter 20, Problem 72

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?

Verified step by step guidance

Understand the problem: The adiabatic index (γ) is defined as the ratio of specific heats, γ = C_p / C_v. For a monatomic gas, γ = 5/3, and for a diatomic gas, γ = 7/5. The goal is to find the fraction of molecules that need to be monatomic in a mixture to achieve an effective γ = 1.50.

Define the variables: Let f be the fraction of molecules that are monatomic, and (1 - f) be the fraction that are diatomic. The effective γ for the mixture can be expressed as a weighted average of the γ values of the two gases: γ_mixture = f * γ_monatomic + (1 - f) * γ_diatomic.

Substitute the known values: γ_monatomic = 5/3, γ_diatomic = 7/5, and γ_mixture = 1.50. The equation becomes: 1.50 = f * (5/3) + (1 - f) * (7/5).

Simplify the equation: Expand and combine terms to isolate f. This involves distributing (1 - f) across (7/5) and then solving for f. The equation becomes: 1.50 = (5/3)f + (7/5) - (7/5)f.

Solve for f: Combine like terms involving f and isolate it on one side of the equation. This will yield the fraction of molecules that need to be monatomic in the mixture to achieve the desired γ value.

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Key Concepts

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

Heat Capacity Ratio (γ)

The heat capacity ratio, denoted as γ (gamma), is the ratio of the specific heat capacity at constant pressure (Cp) to that at constant volume (Cv). For monatomic gases, γ is typically 5/3, while for diatomic gases, it is 7/5. Understanding γ is crucial for analyzing the thermodynamic properties of gases and how they behave under different conditions.

Mixing Gases

When mixing different types of gases, the overall properties of the mixture can be determined by the proportions and individual properties of the gases involved. The effective γ of a gas mixture can be calculated using a weighted average based on the molar fractions of the components, which is essential for solving the problem of achieving a specific γ value through mixing.

Mole Fraction

Mole fraction is a way of expressing the concentration of a component in a mixture, defined as the number of moles of that component divided by the total number of moles of all components. In this context, determining the mole fraction of monatomic and diatomic gases in the mixture is key to achieving the desired γ value of 1.50, as it directly influences the calculated heat capacity ratio of the mixture.

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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}$

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

A 2.0 mol sample of oxygen gas in a rigid, 15 L container is slowly cooled from 250℃ to 50℃ by being in thermal contact with a large bath of 50℃ water. What is the entropy change of (a) the gas and (b) the universe?

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Textbook Question

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.

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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