Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

•• The standard deviation of a random variable x is defined as . Note that this is a measure of the “spread” of values of x about the mean value, (a) Show that the standard deviation can also be written as . (b) What is the standard deviation ∆v of the distribution of speeds of an atom in an ideal monatomic gas at temperature T? (c) What is the ratio ∆v/vrms?