For diatomic carbon dioxide gas (CO2, molar mass 44.044.044.0 g/mol) at T=300T = 300T=300 K, calculate the average speed vavv_{av}.

Ch 18: Thermal Properties of Matter

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 18: Thermal Properties of Matter

Problem 41b

Chapter 18, Problem 41b

For diatomic carbon dioxide gas (CO₂, molar mass $44.0$ g/mol) at $T = 300$ K, calculate the average speed $v_{a v}$ .

Verified step by step guidance

First, understand that the average speed of gas molecules can be calculated using the formula for the average speed $ v_{av} $ of molecules in a gas, which is derived from the Maxwell-Boltzmann distribution. The formula is $ v_{av} = \sqrt{\frac{8kT}{\pi m}} $, where $ k $ is the Boltzmann constant, $ T $ is the temperature in Kelvin, and $ m $ is the mass of a single molecule.

Next, convert the molar mass of CO2 from grams per mole to kilograms per molecule. Since the molar mass is 44.0 g/mol, convert this to kilograms by dividing by 1000, giving 0.044 kg/mol. Then, divide by Avogadro's number $ N_A = 6.022 \times 10^{23} $ molecules/mol to find the mass of a single molecule: $ m = \frac{0.044}{6.022 \times 10^{23}} $ kg.

Substitute the values into the average speed formula. Use $ k = 1.38 \times 10^{-23} $ J/K for the Boltzmann constant and $ T = 300 $ K for the temperature. The mass $ m $ is calculated from the previous step.

Calculate the expression $ \frac{8kT}{\pi m} $ using the values substituted. This involves multiplying $ 8 \times k \times T $ and dividing by $ \pi \times m $.

Finally, take the square root of the result from the previous step to find the average speed $ v_{av} $. This will give you the average speed of CO2 molecules at 300 K.

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Key Concepts

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

Kinetic Theory of Gases

The kinetic theory of gases explains the behavior of gases in terms of the motion of their molecules. It assumes that gas molecules are in constant random motion, and their average kinetic energy is proportional to the temperature of the gas. This theory is essential for calculating properties like average speed, as it relates temperature to molecular motion.

Molar Mass

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). For CO2, the molar mass is 44.0 g/mol, which is crucial for determining the molecular speed. It allows us to convert between the mass of individual molecules and the macroscopic mass of the gas sample, facilitating calculations involving molecular speeds.

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution describes the distribution of speeds among molecules in a gas. It provides a statistical means to calculate the average speed of molecules at a given temperature. The average speed can be derived from this distribution, which is essential for understanding how temperature affects molecular motion in gases like CO2.

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