The ionization temperature of helium atom

[Boltzmann2012]

Homework Statement

It is known that the ionization temperature of hydrogen is T. Then, estimate the ionization temperature of the helium atom in terms of T. Assume that Bohr's theory is applicable, that is,
All the electrons are occupying hydrogenic levels. By ionization temperature, we mean the temperature at which the atomic completely ionized(loses all its electrons).

Homework Equations

Saha ionization equation

The Attempt at a Solution

I am completely confused. When we are asked to assume that Bohr's theory is valid, then how do we account for the repulsion bw electrons in the same shell?
Also, the Saha ionization eqn is valid for thermal equilibrium in which a fraction of atoms are unionized. Here we are dealing with a single atom.

 

[mark726]

First of all, it is important to note that the ionization temperature of an atom is not a fixed value, but rather a range of temperatures at which the atom can become completely ionized. This means that the value of the ionization temperature for helium will be different from that of hydrogen, but will still be related to hydrogen's ionization temperature.

To estimate the ionization temperature of helium, we can use the Saha ionization equation, which relates the ionization energy of an atom to its temperature and the density of free electrons. In this case, we can assume that the density of free electrons is the same for both hydrogen and helium, as they are both at the same temperature and in thermal equilibrium.

The Saha ionization equation is given by:

n+1/n = 2(Z2/h^2) * (2πmkT/h^2)^3/2 * e^(-Eion/kT)

Where n is the number density of free electrons, Z is the atomic number of the atom, h is Planck's constant, m is the mass of the electron, k is Boltzmann's constant, T is the temperature, and Eion is the ionization energy of the atom.

In this case, we can set the left side of the equation to 1, as we are looking for the temperature at which the atom is completely ionized. We can also assume that the atomic number of helium is 2, and that the ionization energy of helium is 4 times that of hydrogen (as it has 4 times as many electrons).

So, we can rewrite the equation as:

1 = 2(2/h^2) * (2πmkT/h^2)^3/2 * e^(-4Eion/kT)

Simplifying, we get:

1 = 8(πmkT/h^2)^3/2 * e^(-4Eion/kT)

Taking the natural log of both sides, we get:

ln(1) = ln(8) + ln((πmkT/h^2)^3/2) + (-4Eion/kT)

Simplifying further, we get:

-4Eion/kT = ln(8) + (3/2)ln(πmkT/h^2)

Finally, solving for T, we get:

T = (-4Eion/k) / (ln(8) + (3/