24 Chopter & Bolteman Stat this nontrivial lower limit, the integral cannot be c

Question

24 Chopter & Bolteman Stat this nontrivial lower limit, the integral cannot be carried option is to do it numerically, by caleulator or computer. Yn With work in unta pg in numbers at this poist, instructing the computer Problem 6.41 ut it's much cleaner to first change variables to ulation 6.49. The integral then becomes Eas of t sost Wkely s u 1000 m/s, that is, Now it's easy to type the integral it 6.5 Partit where the lower limit is the value of when the nitrogen molecules are moving faster Than 1 oxygen (O-) molecules at room temperature. (1000 m/s)/(422 m/s) so and got an answer of 0,0105 for the probability. Only about For an isolat tity is the tu of the mult For a s Problemn 6.33. Calculate the most probable speed, average speed, and rms speed the quant partition the syste its loga we alre Helm we wa there 6.34. Carefully plot the Maxwell speed distribution for K and at T- 600 K. Plot both graphs on the same axes, Ad cules at 35) verify from the Maxwell speed distribution that the tim Problem 6.36. Fill in the steps between equations 6.51 and 6.52, to deternize the average speed of the molecules in an ideal gas. Problem 6.37. Use the Maxwell distribution to calculate the average value of a for the molecules in an ideal gas. Check that your answer agrees with equation 6.41 6.38. At room temperature, what fraction of the nitrogen molecules in e moving at less than 300 m/s? the Problem 6.39. A particle near earth's surface traveling faster than about 11 km/s has enough kinetic energy to completely escape from the earth, despite earth's gravitational pull. Molecules in the upper atmosphere that are moving faster than this will therefore escape if they do not suffer any collisions on the way out. (a) The temperature of earth's upper atmosphere is actually quite high, around 1000 K. Calculate the probability of a nitrogen molecule at this temperature moving faster than 11 km/s, and comment on the result. (b) Repeat the calculation for a hydrogen molecule (H2) and for a helium atom., and discuss the implications. (c) Escape speed from the moon's surface is only about 2.4 km/s. Explain why the moon has no atmosphere. Problem 6.40. You might wonder why all the molecules in a gas in thermal equilibrium don't have exactly the same speed. After all, when two molecules collide, doesn't the faster one always lose energy and the slower one gain energy? 1nd if so, wouldn't repeated collisions eventually bring all the molecules to some In of a hilliard-ball collision in which this is not

Explanation / Answer

Apply:

P(v<300) = 4PI*(m/(2PI*kT)^1.5 * integral from 0 to 300 (V^2 * exp(-mv^2/(2KT)) dv

m = mass of particle, k constnat of Boltzmann, T = Temperature

then

let us assume

x = vsqrt(m/(2KT)

therefore:

4/sqrt(PI) = integral(0-300) * (x^2 * exp(-x^2)) dx

Max spped --> 422 m/s

then V = 300 m

ratio = 300/422 = 0.71

appl yintegral

P(V>300) = 4/sqrt(PI) * integral = 0.71 to 1 *(x^2 * exp(-x^2) dx

P(v> 300) = 4/sqrt(PI) * (0.3511)

P(v > 300) = 4/sqrt(PI) * (0.3511) = 0.789

then

inverse probability --> 1-0.789 = 0.21

fraction will be 0.21 or 21%

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