16.4 Gravitational Effects on the Ideal Gas Model

Overview

The standard Ideal Gas Model assumes that gas molecules interact only through brief elastic collisions and that all other forces — including gravity — are negligible. This assumption holds well at laboratory scales but breaks down over large vertical distances, such as in planetary atmospheres or stellar interiors.

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Why Gravity is Neglected at Laboratory Scale

For a typical laboratory container (height $h\sim 0.1\text{ m}$), the gravitational potential energy change for a single molecule of mass $m$ is:

$\Delta U=mgh$

For a nitrogen molecule ($m\approx 4.65\times 10^{-26}\text{ kg}$) at room temperature:

$\Delta U\approx (4.65\times 10^{-26})(9.8)(0.1)\approx 4.6\times 10^{-26}\text{ J}$

The average thermal kinetic energy at $T=300\text{ K}$ is:

$⟨KE⟩=\frac{3}{2}{k}_{B}T\approx \frac{3}{2}(1.38\times 10^{-23})(300)\approx 6.2\times 10^{-21}\text{ J}$

Since $⟨KE⟩\gg \Delta U$ by a factor of $\sim 10^5$, gravity is completely negligible at laboratory scale. The Ideal Gas Model remains valid.

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Gravitational Effects at Large Scale: The Atmosphere

Over large vertical distances — such as Earth's atmosphere — gravity produces a measurable pressure gradient. As altitude increases:

• The weight of the gas column above decreases
• Both pressure and molecular number density decrease
• The distribution of molecules becomes non-uniform

This is the physical basis for why the air is "thinner" at high altitudes.

The Barometric Formula

For an ideal gas in hydrostatic equilibrium at constant temperature $T$, the pressure at height $h$ is given by the Barometric Formula:

$P(h)=P_0{e}^{-mgh/k_{B}T}$

where:

• $P_0$ = pressure at $h=0$ (sea level)
• $m$ = mass of one gas molecule
• $g$ = gravitational acceleration
• $k_B$ = Boltzmann constant
• $T$ = absolute temperature (assumed constant)

This exponential decrease arises because the ideal gas model, extended to include gravity, predicts that the probability of finding a molecule at height $h$ follows a Boltzmann distribution $e^{-U/k_{B}T}$ where $U=mgh$.

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Gravitational Effects in Stellar Interiors

Inside a star, the gas is subject to enormous gravitational forces due to the star's large mass. The ideal gas model must be extended to account for:

1. Pressure gradient: Pressure increases with depth toward the core
2. Hydrostatic equilibrium: The star is stable when the inward gravitational force is exactly balanced by the outward pressure (thermal pressure from nuclear fusion, radiation pressure, and — in compact objects — degeneracy pressure)

Hydrostatic Equilibrium

For a thin shell of gas at radius $r$ inside a star, the condition for hydrostatic equilibrium is:

$\frac{dP}{dr}=-\rho (r)g(r)$

where $\rho (r)$ is the local density and $g(r)$ is the local gravitational acceleration. This equation shows that the pressure must decrease outward to support the weight of the overlying layers.

Dependence on Mass and Radius

The strength of the inward gravitational pull on a celestial body depends on:

• Its total mass $M$ (greater mass → stronger gravity)
• Its radius $R$ (smaller radius → stronger surface gravity, since $g=GM/R^2$)

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Ideal Gas Model as a Foundation for Statistical Mechanics

The treatment of gravitational effects on an ideal gas illustrates a key principle: the Ideal Gas Model is not merely a simplified approximation — it is the foundation of statistical mechanics. By applying statistical distributions (such as the Boltzmann distribution) to the ideal gas, physicists can:

• Derive the Barometric Formula from first principles
• Understand pressure gradients in atmospheres and stars
• Extend the model to more complex systems (real gases, degenerate matter)

The ideal gas model thus serves as the starting point from which statistical mechanics builds toward understanding matter under all conditions — from everyday gases to extreme astrophysical environments.

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Summary