For a gas of nitrogen molecules (N2), what must the temperature be if 94.7%94.7\% of all the molecules have speeds less than 15001500 m/s? Use Table 18.218.2. The molar mass of N2 is 28.028.0 g/mol.

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 42a

18장, 문제 42a

For a gas of nitrogen molecules (N₂), what must the temperature be if $94.7 percent sign$ of all the molecules have speeds less than $1500$ m/s? Use Table $18.2$ . The molar mass of N₂ is $28.0$ g/mol.

검증된 단계별 안내

First, understand that the problem involves the Maxwell-Boltzmann distribution of molecular speeds in a gas. This distribution describes the probability of molecules having certain speeds at a given temperature.

Next, convert the molar mass of nitrogen (N2) from grams per mole to kilograms per mole. Since 1 gram = 0.001 kilograms, the molar mass of N2 is 28.0 g/mol, which is equivalent to 0.028 kg/mol.

Use the Maxwell-Boltzmann speed distribution formula to relate the fraction of molecules with speeds less than a given value to the temperature. The formula involves the most probable speed, which is given by:

\sqrt{\frac{2 k T}{m}}

, where

k

is the Boltzmann constant,

T

is the temperature, and

m

is the mass of a molecule.

Determine the fraction of molecules with speeds less than 1500 m/s using the cumulative distribution function (CDF) of the Maxwell-Boltzmann distribution. This involves integrating the probability density function up to the speed of 1500 m/s.

Finally, solve for the temperature

T

that corresponds to 94.7% of molecules having speeds less than 1500 m/s. This requires using the inverse of the CDF and the known values to find

T

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주요 개념

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Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution describes the distribution of speeds among molecules in a gas. It is crucial for understanding how molecular speeds vary with temperature. The distribution is characterized by a peak at the most probable speed, and the shape of the curve changes with temperature, affecting the fraction of molecules below a certain speed.

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 Maxwell-Boltzmann distribution. It is calculated using the formula v_rms = sqrt(3kT/m), where k is the Boltzmann constant, T is the temperature, and m is the mass of a molecule. This concept helps relate temperature to molecular speed.

Molar Mass and Molecular Speed

Molar mass is the mass of one mole of a substance, and it plays a critical role in determining the speed of gas molecules. For nitrogen (N2), the molar mass is 28.0 g/mol, which must be converted to kg for calculations. The molecular speed is inversely related to the molar mass, affecting how temperature influences the speed distribution of the molecules.

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