Energy Distribution: 17 flashcards
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1. What is the physical meaning of the Maxwell-Boltzmann energy distribution function $f(E)$?
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It gives the probability density that a particle in a classical ideal gas has energy $E$ at thermal equilibrium. The probability of finding a particle with energy between $E$ and $E+dE$ is $f(E)\,dE$.
2. Write the Maxwell-Boltzmann energy distribution function for a 3D ideal gas.
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$$f(E) = \frac{2}{\sqrt{\pi}} \frac{1}{(k_B T)^{3/2}} \sqrt{E} \, e^{-E/(k_B T)}$$
3. Why does the Maxwell-Boltzmann distribution have a $\sqrt{E}$ factor?
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It arises from the density of states in 3D: the number of states with energy between $E$ and $E+dE$ is proportional to $\sqrt{E}\,dE$ (due to the volume of a sphere in momentum space).
4. What is the most probable energy $E_p$ for a particle in a Maxwell-Boltzmann distribution?
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$$E_p = \frac{1}{2} k_B T$$
5. How does the most probable energy $E_p$ compare to the average energy $\langle E \rangle$?
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$\langle E \rangle = \frac{3}{2} k_B T$, which is three times larger than $E_p = \frac{1}{2} k_B T$.
6. What is the root-mean-square (rms) speed of a molecule in terms of temperature and mass?
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$$v_{\text{rms}} = \sqrt{\frac{3 k_B T}{m}}$$
7. How does the Maxwell-Boltzmann distribution change with increasing temperature?
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The peak shifts to higher energies, the distribution broadens, and the fraction of high-energy particles increases. The area under the curve remains 1 (normalization).
8. What is the normalization condition for the Maxwell-Boltzmann energy distribution?
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$$\int_0^\infty f(E)\,dE = 1$$
9. Derive the average energy $\langle E \rangle$ using the Maxwell-Boltzmann distribution.
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$$\langle E \rangle = \int_0^\infty E \, f(E)\,dE = \frac{3}{2} k_B T$$
10. What is the most probable speed $v_p$ for a molecule in a Maxwell-Boltzmann speed distribution?
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$$v_p = \sqrt{\frac{2 k_B T}{m}}$$
11. How do $v_p$, $\bar{v}$, and $v_{\text{rms}}$ compare for a Maxwell-Boltzmann gas?
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$$v_p < \bar{v} < v_{\text{rms}}$$ with $\bar{v} = \sqrt{\frac{8 k_B T}{\pi m}}$ and $v_{\text{rms}} = \sqrt{\frac{3 k_B T}{m}}$.
12. What is the condition for the Maxwell-Boltzmann distribution to be valid?
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The gas must be dilute (low density) and at sufficiently high temperature so that quantum effects are negligible, i.e., the de Broglie wavelength is much smaller than interparticle spacing.
13. Given a gas at temperature $T$, what fraction of particles have energy greater than $E_0$?
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$$P(E > E_0) = \int_{E_0}^\infty f(E)\,dE$$ which for large $E_0$ approximates to $e^{-E_0/(k_B T)}$ times a polynomial factor.
14. How does the Maxwell-Boltzmann distribution relate to the Boltzmann factor?
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The energy distribution is proportional to the Boltzmann factor $e^{-E/(k_B T)}$ multiplied by the density of states $g(E) \propto \sqrt{E}$.
15. What is the difference between the Maxwell-Boltzmann energy distribution and the speed distribution?
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The energy distribution $f(E)$ gives probability per unit energy, while the speed distribution $f(v)$ gives probability per unit speed. They are related by $f(E)\,dE = f(v)\,dv$ using $E = \frac{1}{2}mv^2$.
16. Calculate the most probable energy for a gas at $T = 300$ K. (Use $k_B = 1.38 \times 10^{-23}$ J/K)
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$$E_p = \frac{1}{2} k_B T = \frac{1}{2} (1.38 \times 10^{-23})(300) \approx 2.07 \times 10^{-21} \text{ J}$$
17. What is the average kinetic energy of a molecule in a monatomic ideal gas at temperature $T$?
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$$\langle K \rangle = \frac{3}{2} k_B T$$
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