For diatomic carbon dioxide gas (CO2, molar mass 44.044.044.0 g/mol) at T=300T = 300T=300 K, calculate the most probable speed vmpv_{mp}.

Ch 18: Thermal Properties of Matter

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 18: Thermal Properties of Matter

문제 41a

18장, 문제 41a

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

검증된 단계별 안내

First, understand that the most probable speed $ v_{mp} $ for a gas molecule is given by the formula $ v_{mp} = \sqrt{\frac{2kT}{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.

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 kg/mol by dividing by 1000, resulting in 0.044 kg/mol. Then, divide by Avogadro's number $ (6.022 \times 10^{23} \text{ molecules/mol}) $ to find the mass of a single molecule.

Substitute the values into the formula for $ v_{mp} $. Use $ T = 300 \text{ K} $ and the calculated mass of a single CO2 molecule.

Calculate the expression $ \frac{2kT}{m} $ using the values for $ k $, $ T $, and $ m $.

Finally, take the square root of the result from the previous step to find the most probable speed $ v_{mp} $.

비슷한 문제에 대한 검증된 영상 답변:

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영상 길이:

주요 개념

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Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution describes the distribution of speeds among particles in a gas. It is crucial for understanding how the most probable speed, average speed, and root-mean-square speed are derived. The distribution is dependent on temperature and particle mass, influencing the speed of gas molecules.

Most Probable Speed

The most probable speed (v_mp) is the speed at which the largest number of gas molecules are moving, according to the Maxwell-Boltzmann distribution. It is calculated using the formula v_mp = sqrt(2kT/m), where k is the Boltzmann constant, T is the temperature, and m is the mass of a molecule.

Molar Mass and Molecular Mass

Molar mass is the mass of one mole of a substance, expressed in grams per mole, and is essential for converting between moles and grams. For gases, the molecular mass is used to calculate the mass of individual molecules, which is necessary for determining speeds using the Maxwell-Boltzmann distribution.