Calculate the fraction of atoms in a sample of argon gas at 400 K that has an energy of 10.0 kJ or greater.

검증된 단계별 안내

Identify the relevant concept: The problem involves calculating the fraction of atoms with a certain energy in a gas, which is related to the Maxwell-Boltzmann distribution.

Use the Maxwell-Boltzmann distribution formula: \( f(E) = \frac{2}{\sqrt{\pi}} \left( \frac{E}{kT} \right)^{1/2} e^{-E/kT} \), where \( E \) is the energy, \( k \) is the Boltzmann constant, and \( T \) is the temperature in Kelvin.

Convert the given energy from kJ to J: Since 1 kJ = 1000 J, convert 10.0 kJ to Joules.

Calculate the exponent in the Maxwell-Boltzmann distribution: \( -E/kT \), where \( k = 1.38 \times 10^{-23} \text{ J/K} \) and \( T = 400 \text{ K} \).

Integrate the Maxwell-Boltzmann distribution from 10.0 kJ to infinity to find the fraction of atoms with energy greater than or equal to 10.0 kJ.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution describes the distribution of speeds (or energies) of particles in a gas at a given temperature. It shows that at higher temperatures, a greater fraction of particles have higher energies. This concept is essential for understanding how many argon atoms will have energies above a certain threshold, such as 10.0 kJ, at 400 K.

Kinetic Energy and Temperature Relationship

The kinetic energy of gas particles is directly related to the temperature of the gas, as described by the equation KE = (3/2)kT for monatomic gases, where k is the Boltzmann constant and T is the temperature in Kelvin. This relationship helps in calculating the average energy of argon atoms at 400 K and determining the fraction that meets or exceeds the specified energy level.

Boltzmann Factor

The Boltzmann factor, given by e^(-E/kT), quantifies the probability of a particle having energy E at temperature T. It is crucial for calculating the fraction of argon atoms with energies equal to or greater than 10.0 kJ. By applying this factor, one can derive the proportion of atoms that possess sufficient energy to exceed the given threshold.