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class11lecnotes - Class 11, Monday, February 1,2010...

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Class 11, Monday, February 1,2010 Reading: 17.1 (The Kinetic Theory of Gases) J f. 2 q {, - z q j- J! ] I The Connection between the Macroscopic and the Microscopic World (Continued) How does the distribution change, if the gas is made of more massive particles? Increasing m p ' while we keep T fixed, leads to a decrease in the most probable speed. Heavier particles will move more slowly, on average, for a given energy (temperature). We should thus expect the curves to shift toward the left when we pick heavier gases. The figure below illustrates this behavior for He, Ne, Ar and Xe. Maxwell-Boltzmann Molecular Speed Distribution for Noble Gases -. -He-4 Ne-20 -Ar-4Q -Xe-132 ~ o 500 1000 1500 2000 2500 Speed (m/s) The dependence of the Maxwell-Boltzmann distribution on both temperature and the mass of the particles has direct implications for a planet's (or moon's) ability to retain an atmosphere. Por a given temperature and gas species, one can compute the root-mean square-speed. If this speed is larger than the escape speed from the planet, then that particular species of gas will be lost over time. Example 1 We can compute this for N z molecules at the current average temperature of the Earth (at the surface), 287K. /?i:T Ir-::-, J rJ -2J1 J 01.';;' I l/;i~! ~ V#;!- ~: !~_~~~~~ c:' j?J1 M,1- o.Jbj~ fY1 t 4, -!' /? . kJ According to this very simplistic consideration, it appears as though ~ cannot escape. Atmospheric escape, however, does not happen at the surface, but high up in] the atmosphere, in a region called the exosphere. The temperature there can be larger t~ ~ factor of~ with respect to the surface. //1 i)ri"/~ ,V;~:"'s 1.e51v-./f - (y, / 7 / it - 1,'1 '.;) ", r '" I
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This is still not enough to push it above Vesc.The escaping particles, however, are the fastest particles in the Maxwell distribution, those in the tail of the distribution. These are speeds ~~1~ larger than the rms-speed. The lower boundary of the exosphere is the exobase. Its location is defined as the altitude at which the mean-free-path of the molecules equals the scale height H (as defined in class 9). Physically, the atmosphere there is so thinned out that the rate of collisions between molecules is greatly reduced. As a result, a given molecule has an increased chance to continue to move upward and escape, if it has enough energy, rather than hit another particle and be stalled in its upward motion. M ~ 1'>1 tJ it uJ. .l/) Example _ Calculating the most probable speed, the average speed, and the rms-speed for air molecules at room temperature and atmospheric escape. ~' ?l: ffvt :: ',:jl''jo II- I'f; :: ,/ 1/. L
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How long would it take a N2 molecule to cross the length of the classroom (about 10m)? This is a random-walk:
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class11lecnotes - Class 11, Monday, February 1,2010...

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