An ammonia bottle is opened very briefly in the center of a large room, releasing many ammonia molecules into the air. These ammonia molecules go on average 10−5 m between collisions with other molecules, and they collide on average 107 times per second. After each collision they are equally likely to go in any direction.
(a) What is the average displacement in one dimension (say the z-dimension) for a single step?
(b) What is the square of the standard deviation for any one step?
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(c) What is the average displacement in the z direction of the escaped ammonia molecules after 2 seconds?
(d) What is the standard deviation of the value obtained in part (c)?
(e) If you were on the z-axis and 6 m from the bottle, how long would it take before more than 32% of the ammonia molecules had positions that were farther from the bottle in the z direction than you?