Questions and Answers
Match the plasma properties with their corresponding equations:
Density of plasma = $\rho = \frac{N}{V}m$ Pressure of plasma = $P = \frac{\rho kT}{m_p + m_e}$ Temperature in the core of the Sun = $T = \frac{m_p P}{2kM}\frac{4}{3}\pi R^3$ Average mass of particles = $m = \frac{m_p + m_e}{2}$
Match the constants with their usage in plasma equations:
k = Boltzmann constant m_p = Mass of a proton m_e = Mass of an electron M = Mass of the Sun
Match the variables with their corresponding units:
P = Pascals T = Kelvin m_p = Kilograms V = Cubic meters
Match the plasma components with their relative abundance in the Sun:
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Match the temperature values with their corresponding descriptions:
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Match the mathematical symbols with their usage in plasma equations:
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Match the following characteristics with the corresponding feature of the degenerate pressure in a plasma:
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Match the following expressions with their physical quantities:
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Match the following types of stars with their corresponding characteristics:
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Match the following conditions with their corresponding type of gas:
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Match the following values with their corresponding physical quantities:
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Match the following expressions with their physical meanings:
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Match the following types of particles with their corresponding roles in the degenerate pressure:
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Match the following conditions with their corresponding types of stars:
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Match the given equations with their corresponding physical concepts:
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Match the variables with their corresponding physical quantities:
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Match the expressions with their corresponding physical concepts:
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Match the given physical quantities with their corresponding units:
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Match the expressions with their corresponding physical concepts in the context of the Sun:
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Match the variables with their corresponding physical quantities in the context of the Sun:
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Match the equations with their corresponding applications:
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Match the physical quantities with their corresponding effects on the pressure gradient:
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Study Notes
Degenerate Pressure
- For a plasma, which is a highly ionized gas, the degenerate pressure is given by the formula:
P = (ℎ² / (5 * 8π * me)) * (ne^(5/3) / (Am*p))
- The degenerate pressure does not depend on the temperature.
- It depends on the inverse of the mass of the particle, specifically 1/me, which means that electrons are the main contributors to degenerate pressure.
- The pressure depends on the density as ρ^(5/3), leading to completely degenerate stars, like white dwarfs, at high density and low temperature.
Degeneracy Condition
- Electrons in a plasma are degenerate if the density of free electrons (ne) is greater than or equal to the density of quantum states (nQ).
- The condition for degeneracy is:
ne ≥ nQ = (4π / (3 * ℎ²)) * (me * kT)^(3/2)
Example: Sun's Core
- In the Sun's core, the density (ρ) is approximately 150 g/cm³, and the temperature (T) is around 15 MK.
- The density of free electrons (ne) is approximately 4.6 x 10²⁵ cm⁻³, which is less than the density of quantum states (nQ), making the gas non-degenerate.
Equation of State
- The equation of state for an ideal gas is:
PV = NkT
, where N is the total number of particles, k is the Boltzmann constant, and T is the temperature. - This approximation is generally good for non-relativistic plasmas.
Hydrostatic Equilibrium
- The gravity force acting on a small cylinder in a plasma is given by:
dFG = -G * M(r) * dm / r²
- The force due to the pressure unbalance is given by:
dFP = -dP * dA
- At equilibrium, the sum of these forces is zero, leading to the equation of hydrostatic equilibrium:
dP / dr = -G * M(r) * ρ(r) / r²
Core Pressure
- The approximate pressure in the core of the Sun can be calculated using the equation:
PC ≈ (G * M²) / (4 * R)
- This gives a pressure of approximately 10¹⁵ Pa in the Sun's core.
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Description
This quiz deals with the calculation of degenerate pressure in plasmas, a highly ionized gas, using formulas involving density and atomic parameters.