Textbook Question
A mixture of chlorine, hydrogen, and oxygen gas is in a container at STP. Which curve represents oxygen gas? (LO 10.8) (a) Curve (a) (b) Curve (b) (c) Curve (c)
7. Gases
Maxwell-Boltzmann Distribution
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What is the fraction of atoms in a sample of argon gas at 400 K that have an energy of 10.0 kJ or greater, according to the Maxwell-Boltzmann distribution?
A
0.0200
B
0.0012
C
0.0100
D
0.0045
Verified step by step guidance1
Understand that the Maxwell-Boltzmann distribution describes the distribution of speeds (and thus kinetic energies) of particles in a gas. The fraction of particles with a certain energy can be found using this distribution.
The probability of finding a particle with energy greater than a certain value is given by the integral of the Maxwell-Boltzmann distribution from that energy to infinity. However, for practical purposes, this is often approximated using tables or computational tools.
Convert the given energy from kJ to J, since the Boltzmann constant is typically expressed in J/K. So, 10.0 kJ = 10,000 J.
Use the formula for the fraction of particles with energy greater than a certain value: \( f(E) = \int_{E}^{\infty} \frac{2}{\sqrt{\pi}} \left( \frac{E}{kT} \right)^{3/2} e^{-E/kT} dE \), where \( E \) is the energy, \( k \) is the Boltzmann constant (1.38 x 10^-23 J/K), and \( T \) is the temperature in Kelvin.
Substitute the values: \( E = 10,000 \) J, \( T = 400 \) K, and \( k = 1.38 \times 10^{-23} \) J/K into the formula and solve for the fraction. This will give you the fraction of atoms with energy 10.0 kJ or greater.
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