Atmospheric Escape and Maxwell-Boltzmann Statistical Mechanics

Atmospheric Escape and the Absence of Diatomic Hydrogen

• Gravitational Retention of Atmosphere

  • The Earth's atmosphere is kept connected to the planet primarily by gravity.

  • Gravity acts as a downward-pulling weight vector that prevents gas molecules from simply floating away into space.

• Factors Influencing Atmospheric Escape

  • In the upper atmosphere, approximately at altitudes of $90\,km$ (the region where space shuttles often operate), temperatures can reach extremes well into the thousands of Kelvin.

  • The escape of molecules depends on the relationship between their kinetic energy (driven by temperature) and the gravitational pull of the Earth.

  • A molecule can escape the atmosphere if its velocity is greater than the escape velocity ($v_e$). The escape velocity condition derived from physics is related to the gravitational constant and the physical properties of the planet:

    • $v_e = \sqrt{\frac{2 \times G \times M}{R}}$

    • $G$ (Gravitational Constant) is on the order of $1 \times 10^{-11}$.

    • $M$ represents the mass of the Earth (which is a massive value).

    • $R$ represents the radius of the Earth.

• The Case of Diatomic Hydrogen ($H_2$)

  • The Root Mean Square velocity ($v_{rms}$) of a molecule is given by the formula:         $v_{rms} = \sqrt{\frac{3k_B T}{m}}$

  • Conditions for escape are met if the temperature ($T$) is high enough and the mass ($m$) of the molecule is small enough.

  • Because $H_2$ has a very low mass, its velocity at high temperatures can exceed the escape threshold defined by the Earth's gravity.

  • Over millions of years, this has caused hydrogen to "bleed out" of our atmosphere, leading to the relative absence of diatomic hydrogen on Earth today.

The Maxwell-Boltzmann Distribution

• Definition and Randomness

  • Ideal gas molecules in a system (like a room) have a random distribution of speeds.

  • While individual velocities are random, they follow a specific statistical distribution known as the Maxwell-Boltzmann distribution.

  • This distribution indicates that certain velocities are more probable (preferential) than others.

• The Probability Equation

  • The probability (frequency) of finding a molecule at a specific speed ($v$) is defined by the following equation:         $f(v) = 4\pi \left( \frac{m}{2\pi k_B T} \right)^{\frac{3}{2}} v^2 e^{-\frac{mv^2}{2k_B T}}$

  • Key Variables include:

    • $m$: Mass of a single molecule.

    • $k_B$: Boltzmann's constant.

    • $T$: Temperature in Kelvin.

    • $v$: Speed (note: speed is always positive, so the horizontal axis of the distribution graph represents speed rather than vectors).

• Graphical Representation and Temperature Effects

  • Cold Gas: Represented by a curve that is taller and narrower, shifted toward the left (lower speeds). The probability of finding molecules at low speeds is high.

  • Hot Gas: As thermal energy increases, the curve shifts to the right (higher speeds) and flattens out (the peak becomes lower).

  • The "Gaussian" part of the function ($e^{-\frac{mv^2}{2k_B T}}$) and the velocity-squared component ($v^2$) combine to create a stretched bell-curve shape.

Statistical Speeds: Most Probable, Average, and RMS

• The Three Key Speed Metrics

  • There are three specific speed values used to describe a gas distribution, which appear in the following order from lowest to highest on the graph:

    1. Most Probable Speed ($v_{mp}$): The speed at the very peak of the distribution; the velocity a molecule is most likely to have.

    2. Average Speed ($v_{avg}$): The mean speed of all molecules, located slightly to the right of the peak.

    3. Root Mean Square Speed ($v_{rms}$): The speed associated with the average kinetic energy, located furthest to the right.

• Speed Formulas (Molar and Molecular)

  • Most Probable Speed:         $v_{mp} = \sqrt{\frac{2RT}{M}} = \sqrt{\frac{2k_B T}{m}}$

  • Average Speed:         $v_{avg} = \sqrt{\frac{8RT}{\pi M}} = \sqrt{\frac{8k_B T}{\pi m}}$

  • RMS Speed:         $v_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3k_B T}{m}}$

• Mathematical Sequence

  • The relative sizes of these speeds are determined by the scalar factors inside the square roots:

    • $2$ (for Most Probable)

    • $\frac{8}{\pi} \approx 2.55$ (for Average)

    • $3$ (for RMS)

  • Since 2 < 2.55 < 3, the order remains consistently v_{mp} < v_{avg} < v_{rms}.

Energy Transfer in Multi-Part Systems

• System Configuration

  • Consider an isolated box containing two sub-systems, Block A and Block B, made of the same material (e.g., gold or a diatomic gas).

  • Sub-systems may have different amounts of matter: $N_A$ (molecules in A) vs. $N_B$ (molecules in B).

  • Initial temperatures differ: T_A > T_B.

• Thermal Equilibrium and Conservation of Energy

  • Heat flows from System A to System B until thermal equilibrium is reached, meaning $T_{A, final} = T_{B, final}$.

  • The Total Energy ($E_{tot}$) of the isolated system is conserved:         $E_{tot} = E_{A, initial} + E_{B, initial}$

  • At equilibrium, while temperatures are the same, the total thermal energies ($E_{A, final}$ and $E_{B, final}$) will only be equal if the number of molecules in each block is identical.

• Calculations for Final Thermal Energy

  • The average kinetic energy per molecule ($K_{avg}$) is a function of temperature: $K_{avg} = \frac{3}{2} k_B T$. Since temperatures are equal at equilibrium, the average kinetic energy per molecule is the same for both systems:         $\frac{E_{A, final}}{N_A} = \frac{E_{B, final}}{N_B} = \frac{E_{tot}}{N_A + N_B}$

  • To find the final energy in System A:         $E_{A, final} = N_A \times \left( \frac{E_{tot}}{N_A + N_B} \right)$

  • To find the final energy in System B:         $E_{B, final} = N_B \times \left( \frac{E_{tot}}{N_A + N_B} \right)$

• Application to Moles

  • These equations also work using moles ($n$) because the conversion constants cancel out:         $E_{A, final} = n_A \times \left( \frac{E_{tot}}{n_A + n_B} \right)$         $E_{B, final} = n_B \times \left( \frac{E_{tot}}{n_A + n_B} \right)$

• Special Case: Identical Systems

  • If $n_A = n_B$, the energy splits equally:         $E_{A, final} = n_A \times \left( \frac{E_{tot}}{2n_A} \right) = \frac{1}{2} E_{tot}$

  • This confirms that identical systems at thermal equilibrium share the total system energy 50/50.

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  • Energy transfers must always obey the conservation of energy.

  • This principle is formalised as the First Law of Thermodynamics.