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.

•• Make an order-of-magnitude estimate of the temperature at which the internal vibrational mode of a nitrogen molecule becomes appreciably excited. Recall that the energy of the first excited state of a simple harmonic oscillator is ℏω, where is the classical angular frequency, k is the spring constant, and m is the mass. Recall also that the energy of the oscillator can be written as . Finally, note that we can expect a typical chemical bond energy U(x) ~ 3 eV when x is a typical atomic distance x ≈ 0.1 nm. The moral of this question is that the vibrational modes of air molecules are largely “frozen out” at room temperature.