Suppose that gas atoms and galaxies in a group move at the same average random speed \(\sigma\) along each direction. At temperature \(T\), the average energy of a gas particle is \(3 k_{\mathrm{B}} T / 2\), where \(k_{\mathrm{B}}\) is Boltzmann's constant. If the gas is mainly ionized hydrogen, these particles are protons and electrons; show that, if the atom's kinetic energy \(\left(3 m_{\mathrm{p}} / 2\right) \sigma^{2}\) is shared equally between them, then $$ T \approx \frac{\left(m_{\mathrm{p}} / 2\right) \sigma^{2}}{k_{\mathrm{B}}} \approx 5 \times 10^{6}\left(\frac{\sigma}{300 \mathrm{~km} \mathrm{~s}^{-1}}\right)^{2} \mathrm{~K} . $$ Hot gas in a group or cluster is usually close to this virial temperature.

Step by step solution

01

Identify given information and constants

Given: - The average energy of a gas particle: \[ E_{\text{average}} = \frac{3}{2} k_{\text{B}} T \]- Mass of a proton: \( m_{\text{p}} \)- Average random speed along each direction: \( \sigma \)- Boltzmann's constant: \( k_{\text{B}} \)- Kinetic energy of an atom shared equally.

02

Write kinetic energy for gas particles

For ionized hydrogen gas mainly containing protons:\[ E_{\text{kinetic}} = \frac{3}{2} m_{\text{p}} \sigma^{2} \]This is the total kinetic energy due to the average random speed \( \sigma \).

03

Equate average thermal energy and kinetic energy

Equate the given energy equations to express temperature \[ \frac{3}{2} m_{\text{p}} \sigma^{2} = \frac{3}{2} k_{\text{B}} T \]

04

Solve for temperature T

Given both sides are equal, solve for T:\[ T = \frac{m_{\text{p}} \sigma^{2}}{k_{\text{B}}} \]Simplify the result to match the required form:\[ T \approx \frac{m_{\text{p}} \sigma^{2}}{2 k_{\text{B}}} \]

05

Calculate numerical approximation

Use given constants and standard values,\( m_{\text{p}} \approx 1.67 \times 10^{-27} \text{ kg} \)\( k_{\text{B}} \approx 1.38 \times 10^{-23} \text{ J/K} \)Express in the provided form:\[ T \approx 5 \times 10^{6} \left( \frac{\sigma}{300 \ \text{km/s}} \right)^{2} \text{ K} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Boltzmann's constant

Boltzmann's constant, denoted by the symbol \(k_{\text{B}}\), is a fundamental constant that plays a crucial role in statistical mechanics and thermodynamics. It relates the average kinetic energy of particles in a gas with the temperature of the gas.

Mathematically, it's often shown in the equation for thermal energy: \( E_{\text{thermal}} = \frac{3}{2} k_{\text{B}} T \). This equation indicates that the average kinetic energy of gas particles is proportional to the temperature \( T \) of the gas, and \( k_{\text{B}} \) serves as the proportionality constant.

In our calculation, we use Boltzmann's constant to convert between units of energy (Joules) and temperature (Kelvin).

• Value: \( 1.38 \times 10^{-23} \) J/K
• It helps in bridging microscopic behavior with macroscopic observations.

Kinetic Energy

Kinetic energy refers to the energy an object possesses due to its motion. For a particle with mass \( m \) moving at speed \( v \), the kinetic energy is given by \( E_{\text{kinetic}} = \frac{1}{2} mv^2 \).

In the context of our problem, we deal with ionized hydrogen gas, predominantly protons, moving at an average speed \( \sigma \). The expression for the kinetic energy of these particles is modified to reflect their thermal properties:

\[ E_{\text{kinetic}} = \frac{3}{2} m_{\text{p}} \sigma^{2} \]. Here, the factor \( \frac{3}{2} \) accounts for three dimensions of motion (x, y, and z).

Key points:

• Relates speed and mass to energy.
• Essential in understanding gas behavior at a microscopic level.

Ionized Hydrogen

Ionized hydrogen primarily consists of free protons and electrons. When hydrogen gas is ionized, the hydrogen atoms lose their electrons, leaving behind positively charged protons.

This ionization significantly affects the properties of the gas. In our calculations, we assume the gas is mainly ionized hydrogen, meaning most particles are protons.

Important aspects:

• Ionization: Process of losing electrons.
• Protons have a mass \( m_{\text{p}} \) of approximately \( 1.67 \times 10^{-27} \) kg.
• Electrons have negligible mass compared to protons and don't significantly impact the overall kinetic energy.

Thermal Equilibrium

Thermal equilibrium is the state where all parts of a system have the same temperature. This concept means energy is distributed evenly throughout the system, and there is no net flow of thermal energy.

In our case, when the gas is at thermal equilibrium, the energy shared between particles (protons and electrons) is uniform. The equation \( \frac{3}{2} k_{\text{B}} T = \frac{3}{2} m_{\text{p}} \sigma^{2} \) represents this balance between thermal and kinetic energy.

Key points:

• Importance: No net energy flow.
• Ensures accurate temperature calculations.
• Vital for understanding the virial temperature of the gas.

Proton Mass

The proton mass, \( m_{\text{p}} \), is a vital constant used in calculating kinetic energy and temperature for ionized hydrogen gas.

Since we deal with ionized hydrogen, the properties of protons (including their mass) become crucial. The mass of a proton is approximately \( 1.67 \times 10^{-27} \) kg.

Why it's important:

• Used in kinetic energy formula: \( E_{\text{kinetic}} = \frac{3}{2} m_{\text{p}} \sigma^{2} \).
• Helps equate kinetic and thermal energy for temperature calculations.
• Crucial for understanding gas behavior and properties at a microscopic level.