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MMLU-Pro / 9771 / A crude model for the molecular distribution of atmospheric gases above Earth's…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

question

A crude model for the molecular distribution of atmospheric gases above Earth's surface (denoted by height hh ) can be obtained by considering the potential energy due to gravity: P(h)=e−mgh/kT P(h)=e^{-m g h / k T} In this expression mm is the per-particle mass of the gas, gg is the acceleration due to gravity, kk is a constant equal to $1.38 \times 10^{-23} \mathrm{~J} \mathrm{~K}^{-1}$, and TT is temperature. Determine ⟨h⟩\langle h\rangle for methane (CH4)\left(\mathrm{CH}_4\right) using this distribution function.
Plain-text mathematical notation (without MathML)
A crude model for the molecular distribution of atmospheric gases above Earth's surface (denoted by height h ) can be obtained by considering the potential energy due to gravity:
P(h)=e^(−mgh/kT)
In this expression m is the per-particle mass of the gas, g is the acceleration due to gravity, k is a constant equal to $1.38 \times 10^{-23} \mathrm{~J} \mathrm{~K}^{-1}$, and T is temperature. Determine ⟨h⟩ for methane (CH₄) using this distribution function.
Original LaTeX notation
A crude model for the molecular distribution of atmospheric gases above Earth's surface (denoted by height $h$ ) can be obtained by considering the potential energy due to gravity:
$$
P(h)=e^{-m g h / k T}
$$
In this expression $m$ is the per-particle mass of the gas, $g$ is the acceleration due to gravity, $k$ is a constant equal to $1.38 \times 10^{-23} \mathrm{~J} \mathrm{~K}^{-1}$, and $T$ is temperature. Determine $\langle h\rangle$ for methane $\left(\mathrm{CH}_4\right)$ using this distribution function.

category

physics

options

  1. 1.2 $10^4 \mathrm{~m}$
  2. 2.5 $10^4 \mathrm{~m}$
  3. 1.8 x 10^4 m
  4. 1.0 x 10^4 m
  5. 1.4 x 10^4 m
  6. 1.6 $10^4 \mathrm{~m}$
  7. 2.0 $10^4 \mathrm{~m}$
  8. 3.5 x 10^4 m
  9. 3.0 x 10^4 m
  10. 2.2 x 10^4 m

src

scibench-thermo

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Source and history

upstream_id / Original problem ID
9771
task_id
c99b9d03-1376-5985-9a4f-58a980a96606
task_key
default--test--9771
task_revision_id
1
config / split
default / test

Official source

initial import