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The energy required to ionise a hydrogen atom is 13.6 eV, so you might expect that the number of ionised hydrogen atoms in the sun's atmosphere would be even less than the number in the first excited state. Yet at the end of Chapter 5 I showed that the fraction of ionised hydrogen is much larger, nearly one atom in 10,000. Explain why this result is not a contradiction, and why it would be incorrect to try to calculate the fraction of ionised hydrogen using the methods of this section.

Short Answer

Expert verified

It makes estimating the probabilities unknown, it does make the ionisation state more likely than using solely Boltzmann factors.

Step by step solution

01

Given information

The energy required to ionise a hydrogen atom is 13.6 eV, so you might expect that the number of ionised hydrogen atoms in the sun's atmosphere would be even less than the number in the first excited state. Yet at the end of Chapter 5 I showed that the fraction of ionised hydrogen is much larger, nearly one atom in 10,000.

02

Explanation

The probability of ionised atoms is equal to Boltzmann factors e-I/kTwhere I is the ionisation energy and T is the temperature, plus the Boltzmann factors multiplied by the degeneracy of the ionised state, which is essentially limitless. Although this makes estimating the probabilities unknown, it does make the ionisation state more likely than using solely Boltzmann factors. The likelihood does, in fact, depend on the number density of electrons in the surroundings.

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Most popular questions from this chapter

Consider a large system of Nindistinguishable, noninteracting molecules (perhaps an ideal gas or a dilute solution). Find an expression for the Helmholtz free energy of this system, in terms of Z1, the partition function for a single molecule. (Use Stirling's approximation to eliminate the N!.) Then use your result to find the chemical potential, again in terms ofZ1.

Consider a classical "degree of freedom" that is linear rather than quadratic E=cqfor some constant c. (As example would be the kinetic energy of a highly relativistic particle in one dimension, written in terms of its momentum.) Repeat derivation of the equipartition theorem for this system, and show that the average energy isrole="math" localid="1646903677918" E-=kT.

Prove that, for any system in equilibrium with a reservoir at temperature T, the average value of E2 is

E2¯=1Z2Zβ2

Then use this result and the results of the previous two problems to derive a formula for σEin terms of the heat capacity, C=E¯/T

You should findσE=kTC/k

At room temperature, what fraction of the nitrogen molecules in the air are moving at less than300m/s?

Imagine a particle that can be in only three states, with energies -0.05 eV, 0, and 0.05 eV. This particle is in equilibrium with a reservoir at 300 K.

(a) Calculate the partition function for this particle.

(b) Calculate the probability for this particle to be in each of the three states.

(c) Because the zero point for measuring energies is arbitrary, we could just as well say that the energies of the three states are 0, +0.05 eV, and +0.10 eV, respectively. Repeat parts (a) and (b) using these numbers. Explain what changes and what doesn't.

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