Ch 18: Thermal Properties of Matter

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 18: Thermal Properties of Matter

Problem 32a

Chapter 18, Problem 32a

The atmosphere of Mars is mostly CO₂ (molar mass $44.0$ g/mol) under a pressure of $650$ Pa, which we shall assume remains constant. In many places the temperature varies from $0.0$ °C in summer to $negative 100$ °C in winter. Over the course of a Martian year, what are the ranges of the rms speeds of the CO₂ molecules.

Verified step by step guidance

First, convert the given temperatures from Celsius to Kelvin. The formula for conversion is: $ T(K) = T(°C) + 273.15 $. So, for summer, $ T_{summer} = 0.0°C + 273.15 = 273.15 \text{ K} $ and for winter, $ T_{winter} = -100°C + 273.15 = 173.15 \text{ K} $.

Next, use the formula for the root mean square (rms) speed of gas molecules: $ v_{rms} = \sqrt{\frac{3kT}{m}} $, where $ k $ is the Boltzmann constant $ (1.38 \times 10^{-23} \text{ J/K}) $, $ T $ is the temperature in Kelvin, and $ m $ is the mass of a single molecule in kilograms.

Calculate the mass of a single CO2 molecule. The molar mass of CO2 is 44.0 g/mol, which is $ 44.0 \times 10^{-3} \text{ kg/mol} $. To find the mass of one molecule, divide by Avogadro's number $ (6.022 \times 10^{23} \text{ molecules/mol}) $: $ m = \frac{44.0 \times 10^{-3}}{6.022 \times 10^{23}} \text{ kg} $.

Substitute the values for summer into the rms speed formula: $ v_{rms, summer} = \sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 273.15}{m}} $.

Substitute the values for winter into the rms speed formula: $ v_{rms, winter} = \sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 173.15}{m}} $. This will give you the range of rms speeds over the course of a Martian year.

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Key Concepts

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

Root Mean Square Speed

The root mean square (rms) speed is a measure of the average speed of particles in a gas, derived from the kinetic theory of gases. It is calculated using the formula v_rms = sqrt((3kT)/m), where k is the Boltzmann constant, T is the temperature in Kelvin, and m is the molar mass of the gas. This concept helps in understanding how temperature affects the speed of gas molecules.

Temperature Conversion

Temperature conversion is essential for calculations involving gas laws, as these typically require temperatures in Kelvin. To convert Celsius to Kelvin, add 273.15 to the Celsius temperature. For Mars, the temperatures of 0.0°C and -100°C convert to 273.15 K and 173.15 K, respectively, which are necessary for calculating the rms speeds of CO2 molecules.

Ideal Gas Law

The ideal gas law, PV = nRT, relates the pressure, volume, and temperature of a gas with its amount in moles. Although the pressure is constant in this scenario, understanding this law helps in comprehending how temperature variations affect gas properties. It provides a framework for analyzing the behavior of gases under different conditions, such as those on Mars.

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Related Practice

Textbook Question

A flask contains a mixture of neon (Ne), krypton (Kr), and radon (Rn) gases. Compare the root-mean-square speeds. (Hint: Appendix D shows the molar mass (in g/mol) of each element under the chemical symbol for that element.)

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Textbook Question

Oxygen (O₂) has a molar mass of $32.0$ g/mol. What is the average translational kinetic energy of an oxygen molecule at a temperature of $300$ K?

Textbook Question

A flask contains a mixture of neon (Ne), krypton (Kr), and radon (Rn) gases. Compare the average kinetic energies of the three types of atoms.

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Textbook Question

Oxygen (O₂) has a molar mass of $32.0$ g/mol. Suppose an oxygen molecule traveling at this speed bounces back and forth between opposite sides of a cubical vessel $0.10$ m on a side. What is the average force the molecule exerts on one of the walls of the container? (Assume that the molecule's velocity is perpendicular to the two sides that it strikes.)

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Textbook Question

Oxygen (O₂) has a molar mass of $32.0$ g/mol. What is the momentum of an oxygen molecule traveling at this speed?

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Textbook Question

We have two equal-size boxes, A and B. Each box contains gas that behaves as an ideal gas. We insert a thermometer into each box and find that the gas in box A is at $50$ °C while the gas in box B is at $10$ °C. This is all we know about the gas in the boxes. Which of the following statements must be true? Which could be true? Explain your reasoning.

(a) The pressure in A is higher than in B.

(b) There are more molecules in A than in B.

(d) The molecules in A have more average kinetic energy per molecule than those in B.

(e) The molecules in A are moving faster than those in B.

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