- #1
DavideGenoa
- 155
- 5
Hi, friends!
1. Homework Statement :
I have found an exercise where one should calculate the ratio ##N_{v_{\text{e}}}/N_{v_{\text{rms}}}## between the number of molecules having escape speed ##v_{\text{e}}=\sqrt{(2G M_t)/r}## (where ##M## is the mass of Earth) and those moving at the root mean square speed ##v_{\text{rms}}=\sqrt{(3RT)/M}## (where ##M## is the molar mass) in the nitrogen found at a height of ##150 \text{ km}## and a temperature of ##1000\text{ K}##.
I know the Maxwell-Boltzmann distribution of molecular speeds
The values of ##v_{\text{f}}\approx1.1\cdot 10^4\text{ m/s}## and ##v_{\text{qm}}\approx 940\text{ m/s}## are immediately computed, but I have no idea how to find ##N_{v_{\text{e}}}/N_{v_{\text{rms}}}##. I do not know if I can use the Maxwell-Boltzmann distribution nor how to use it, since, with such a continuous distribution, the probability that a molecule has a definite velocity, rather than being in an interval, is zero. I am not able to use, if it is useful in this case, the approximation ##\Delta n\approx N f(v)\Delta v##, either.
I ##\infty##-ly thank you all!
1. Homework Statement :
I have found an exercise where one should calculate the ratio ##N_{v_{\text{e}}}/N_{v_{\text{rms}}}## between the number of molecules having escape speed ##v_{\text{e}}=\sqrt{(2G M_t)/r}## (where ##M## is the mass of Earth) and those moving at the root mean square speed ##v_{\text{rms}}=\sqrt{(3RT)/M}## (where ##M## is the molar mass) in the nitrogen found at a height of ##150 \text{ km}## and a temperature of ##1000\text{ K}##.
Homework Equations
I know the Maxwell-Boltzmann distribution of molecular speeds
##f(v)=4\pi\bigg(\frac{M}{2\pi RT }\bigg)^{\frac{3}{2}}e^{-\frac{M}{2RT}v^2}##
which means that, if we call ##N## the total number of molecules and ##\Delta n## is the number of molecules having the speed in the interval ##\Delta v##,##dn=N f(v)dv.##
The Attempt at a Solution
The values of ##v_{\text{f}}\approx1.1\cdot 10^4\text{ m/s}## and ##v_{\text{qm}}\approx 940\text{ m/s}## are immediately computed, but I have no idea how to find ##N_{v_{\text{e}}}/N_{v_{\text{rms}}}##. I do not know if I can use the Maxwell-Boltzmann distribution nor how to use it, since, with such a continuous distribution, the probability that a molecule has a definite velocity, rather than being in an interval, is zero. I am not able to use, if it is useful in this case, the approximation ##\Delta n\approx N f(v)\Delta v##, either.
I ##\infty##-ly thank you all!