# Solving for degeneracy electron cloud temperature

• kamui1
In summary, To solve the given question about finding the maximum temperature with a given pressure, you will need to use the ideal gas law equation and the equation of state for a real gas. You will need to rearrange the equation of state to solve for T and then substitute the given values to calculate the maximum temperature.
kamui1
Homework Statement
Assume degeneracy starts to dominate approximately when the pressure expected from a completely degenerate electron gas equals that expected from an ideal gas (P_rel_e=P_ideal=1/2Ptotal) Show that the temperature at that point, which we assume will be the maximum reached, is given by T_max is true
Relevant Equations
Tmax= 7.7x10^7 K mu*mu_e^5/3*(M/M_solar)^5/3
P_rel_e = K(ro/mu_e*m_h)^5/3
P_ideal = nKT
When I try P_rel_e = P_ideal I couldn't get a single number that is close to the given T_Max. It might be that I used the wrong equations but I am not sure. Can anyone give me some guidence on this question?

To solve this question, you will need to use the ideal gas law equation and the equation of state for a real gas. The ideal gas law equation is P*V = n*R*T, where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the temperature. The equation of state for a real gas is PV = nRT + a(n2/V2), where a is an empirical constant that depends on the type of gas.To find the maximum temperature with a given pressure, you will need to solve the equation of state for the temperature. First, rearrange the equation of state to solve for T:T = (PV - a(n2/V2))/(nR)Then, substitute the known values of P, V, n, and R into the equation to solve for T. You can then compare the calculated maximum temperature to the given value to see if it is close.

## What is degeneracy in the context of electron clouds?

Degeneracy in the context of electron clouds refers to the number of different quantum states that have the same energy level. In a degenerate system, multiple electrons can occupy states with the same energy, which is significant in systems such as atoms, molecules, and solids where electron configurations play a crucial role in determining physical properties.

## Why is it important to solve for the electron cloud temperature in degenerate systems?

Solving for the electron cloud temperature in degenerate systems is important because it helps in understanding the thermal properties and behaviors of materials, especially at very low temperatures where quantum effects become significant. It is crucial for areas like condensed matter physics, astrophysics, and material science, where electron interactions and thermal dynamics dictate the properties of the system.

## How does one typically solve for the electron cloud temperature in a degenerate system?

To solve for the electron cloud temperature in a degenerate system, one typically uses statistical mechanics and quantum theory. The process involves calculating the Fermi energy and using the Fermi-Dirac distribution to determine the population of electrons at various energy levels. From this, the temperature can be inferred by examining the distribution of electrons and their energy states.

## What role does the Fermi-Dirac distribution play in determining electron cloud temperature?

The Fermi-Dirac distribution is crucial in determining the electron cloud temperature because it describes the probability of electron occupancy in energy states at a given temperature. By analyzing how electrons populate these states, especially close to the Fermi level, scientists can infer the temperature of the electron cloud. This distribution is particularly important in degenerate systems where quantum effects dominate.

## Can external factors like magnetic fields or pressure affect the electron cloud temperature in degenerate systems?

Yes, external factors such as magnetic fields or pressure can significantly affect the electron cloud temperature in degenerate systems. Magnetic fields can alter the energy levels and the distribution of electrons through phenomena like the Zeeman effect. Pressure can change the density and spacing of atoms, thereby affecting the electron energy states and their distribution. Both factors can lead to changes in the electron cloud temperature by modifying the conditions under which electrons interact and occupy energy levels.

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