Question 8.3: Find the value of the dimensionless velocity variable u for ...
Find the value of the dimensionless velocity variable u for a hydrogen atom moving with the escape velocity of the Earth in a region of the thermosphere for which the temperature is 1700 K and find the probability a hydrogen atom in this region of the thermosphere will escape into outer space.
Using Eq. (8.23) and the physical constants in Appendix A, we find
| Appendix A | |||
| Constants and conversion factors | |||
| Constants | |||
| Speed of light | c | 2.99792458 × 10^{8} m/s | |
| Charge of electron | e | 1.6021773 × 10^{−19} C | |
| Plank’s constant | h | 6.626076 × 10^{−34} J s | |
| 4.135670 × 10^{−15} eV s | |||
| \hbar=h/2π | 1.054573 × 10^{−34} J s | ||
| 6.582122 × 10^{−16} eV s | |||
| hc | 1239.8424 eV nm | ||
| 1239.8424 MeV fm | |||
| Hydrogen ionization energy | 13.605698 eV | ||
| Rydberg constant | 1.0972 × 10^{5} cm^{−1} | ||
| Bohr radius | a_{0} = (4π \epsilon _{0})/(me²) | 5.2917725 × 10^{−11} m | |
| Bohr magneton | μ_{B} | 9.2740154 × 10^{−24} J/T | |
| 5.7883826 × 10^{−5} eV/T | |||
| Nuclear magneton | μ_{N} | 5.0507865 × 10^{−27} J/T | |
| 3.1524517 × 10^{−8} eV/T | |||
| Fine structure constant | α = e^{2}/(4π\epsilon _{0} c \hbar) | 1/137.035989 | |
| e^{2}/4π\epsilon _{0} | 1.439965 eV nm | ||
| Boltzmann constant | k | 1.38066 × 10^{−23} J/K | |
| 8.6174 × 10^{−5} eV/K | |||
| Avogadro’s constant | N_{A} | 6.022137 × 10^{23} mole | |
| Stefan-Boltzmann constant | σ | 5.6705 × 10^{−8} W/m² K^{4} | |
| Particle masses | |||
| kg | u | MeV/c² | |
| Electron | 9.1093897 × 10^{−31} | 5.485798 × 10^{−4} | 0.5109991 |
| Proton | 1.6726231 × 10^{−27} | 1.00727647 | 938.2723 |
| Neutron | 1.674955 × 10^{−27} | 1.008664924 | 939.5656 |
| Deuteron | 3.343586 × 10^{−27} | 2.013553 | 1875.6134 |
| Conversion factors | |||
| 1 eV | 1.6021773 × 10^{−19} J | ||
| 1 u | 931.4943 MeV/c² | ||
| 1.6605402 × 10^{−27} kg | |||
| 1 atomic unit | 27.2114 eV | ||
u=v\left(\frac{m}{2\pi k_{B}T} \right)^{1/2} . (8.23)
u =\left(\frac{1.027865 × 1.6605402 × 10^{−27} \ kg}{2 × 1.38066 × 10^{−23} \ J/K × 1700 \ K}\right) ^{1/2} × 11.2 × 10^{3} \ m/s = 2.13564The probability that a hydrogen atom will escape into outer space due to its thermal motion depends upon the probability a hydrogen atom will have a velocity greater than the escape velocity. This probability can be calculated using the following one-line MATLAB ^{\circledR} program that integrates the probability from the value of u we have just calculated to a very large number.
quadl(@(u)u.^2.*exp(-u.^2),2.13564,10)
This program returned the answer u = 0.0123, which means that a hydrogen atom in this region of the thermosphere moving away from our planet has a slightly more than one percent chance of escaping into outer space. The process by which molecules of a planetary atmosphere escape into outer space due to their thermal motion is called thermal escape. As described in the classic book, “Theory of Planetary Atmospheres”, cited at the end of this chapter, atmospheric escape is a complex process that includes many different processes. For example, recombination processes such as O^{+}_{2} + e ⇒ O + O turns the binding energy of the O^{+}_{2} ion into kinetic energy of the product O atoms that can lead to their escape into outer space. Recent attempts to find planets orbiting distant stars has renewed interest in planetary atmospheres.