====== Ideal gas ======

In astrophysics, an **ideal gas** is one whose particles obey the Maxwell–Boltzmann statistics and interact only through elastic collisions, without long-range forces.  
It provides an excellent approximation for the hot, low-density plasma inside stars, where interactions between particles are rare and short-lived.

The **equation of state (EOS)** of an ideal gas expresses the pressure \(P\) as a function of density \(\rho\) and temperature \(T\).

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===== Derivation of pressure =====

Consider a particle moving toward a wall with velocity \(\mathbf{v}\) and momentum \(\mathbf{p} = m\mathbf{v}\).  
Upon an elastic collision, it reverses direction, transferring momentum to the wall:

$$
\Delta \mathbf{p} = \mathbf{p}' - \mathbf{p} = -m\mathbf{v} - m\mathbf{v} = -2m\mathbf{v}.
$$

Hence, \(+2m\mathbf{v}\) of momentum is imparted to the wall in each collision.  
Pressure is defined as the momentum transferred per unit area per unit time:

$$
P = N \frac{\Delta p}{\Delta t} \frac{1}{\Delta A}.
$$

If the total number density of particles is \(n\), then the number hitting a single wall per unit time is \(N = (nV)/6 = (nv\Delta t \Delta A)/6\).  
Only one-sixth of all particles move toward a given wall (since there are six directions: ±x, ±y, ±z).

Substituting, we find:

$$
P = \frac{nv}{6} (2mv) = \frac{1}{3} n m v^2 = \frac{2}{3} n \left(\frac{1}{2} m v^2\right).
$$

This shows that pressure is directly proportional to the **kinetic energy density** of the gas:

$$
P = \frac{2}{3} n E,
$$

where \(E\) is the average kinetic energy per particle.

If the particles have a distribution of speeds (as in Maxwell–Boltzmann statistics), then

$$
P = \frac{2}{3} n E_{av}.
$$

Pressure and kinetic energy density have identical dimensions (energy per volume).  
Although pressure is formally a **tensor** (involving both direction and force), it reduces to a scalar in an isotropic gas, where motion is equally probable in all directions.

===== Average kinetic energy =====

For a gas following the Maxwell–Boltzmann distribution, the mean kinetic energy per particle can be derived as the expected value:

$$
E_{av} = \left(\frac{p^2}{2m}\right)_{av}
= \int_0^\infty \frac{p^2}{2m} P(p)\,dp
= \frac{3}{2} kT.
$$

Thus, temperature \(T\) is a direct measure of the average kinetic energy of the particles.

This relation allows us to infer the **spectral emission** of a gas.  
A monatomic hydrogen gas at \(T = 12\ \text{MK}\) has \(E_{av} = kT = 1.5\ \text{keV}\), corresponding to photons with frequency \(h\nu \approx kT\), or about \(10^{17}\ \text{Hz}\) — in the **X-ray** range.  
At \(T = 6000\ \text{K}\), radiation shifts to the **visible** range, as in the solar photosphere.

===== Degrees of freedom and molecular gases =====

For a **monatomic** gas, motion in the three spatial directions gives **three degrees of freedom (DOF)**.  
Each contributes \(\tfrac{1}{2}kT\) to the energy, yielding

$$
E_{av} = \frac{3}{2} kT.
$$

A **diatomic** gas has two additional DOFs — rotation and vibration — giving

$$
E_{av} = \frac{5}{2} kT.
$$

However, in stellar interiors, molecules dissociate under extreme temperatures, so the gas is almost entirely monatomic and ionized.

===== Equation of state for stellar gas =====

In stellar interiors, pressure arises mainly from **thermal motion** of ions and electrons.  
Using the ideal gas law, the pressure is

$$
P = n kT = \frac{\rho}{m_{av}} kT,
$$

where \(m_{av}\) is the average mass per particle.

For a fully ionized hydrogen plasma, each hydrogen atom contributes two particles (a proton and an electron), so \(m_{av} = m_p/2\).  
This form of the EOS,

$$
P = \frac{\rho kT}{m_{av}},
$$

is fundamental in **stellar structure** equations and serves as the foundation for hydrostatic equilibrium, virial balance, and energy transport analyses.

The more familiar form of the **ideal gas law**, \(PV = \mathsf{n}RT\), follows from \(n = \mathsf{n} N_0 / V\), where \(N_0\) is Avogadro’s number and \(R = k N_0\) is the universal gas constant.

===== Astrophysical context =====

  - In the **solar core**, \(T \sim 1.6 \times 10^7\ \text{K}\) and \(\rho \sim 1.5 \times 10^5\ \text{kg/m}^3\), giving pressures of order \(10^{16}\ \text{Pa}\).
  - The **ideal gas EOS** is valid wherever particle collisions dominate over degeneracy or radiation pressure — i.e., in most main-sequence stars.
  - At higher densities or lower temperatures, quantum corrections become important, leading to **degenerate gases**, such as in white dwarfs and neutron stars.

===== Insights =====
  - Pressure in an ideal gas arises from the transfer of momentum by particle collisions.
  - For an isotropic distribution, \(P = \tfrac{2}{3} nE_{av}\).
  - The average kinetic energy per particle, \(E_{av} = \tfrac{3}{2}kT\), defines temperature microscopically.
  - The stellar equation of state, \(P = \rho kT / m_{av}\), links thermodynamics to hydrostatic equilibrium.
  - Deviations from the ideal gas law occur when degeneracy or radiation pressure dominates.

===== Inquiries =====
  - Derive the expression \(P = \tfrac{1}{3} n m v^2\) from the momentum transfer argument.
  - Show how integrating the Maxwell–Boltzmann distribution leads to \(E_{av} = \tfrac{3}{2} kT\).
  - Why does a fully ionized hydrogen plasma have \(m_{av} = m_p / 2\)?
  - In which astrophysical environments does the ideal gas law fail, and why?
  - How does the ideal gas EOS connect to hydrostatic equilibrium in stars?