Midterm Solution note and guide

1. the equation for kinetic energy is (1/2) m v^2.
[side note: This little equation relates the two parts of this class. It is the equation for kinetic energy for macroscopic masses, and for microscopic atoms (where it is intimately involved in the microscopic definition of temperature.]

2. a) At T=0 K there is no motion; atom motion ceases...
b) Energy is not create or destroyed; it (only) transforms from one form to another.

3 a) He: 2,2,2; Li: 3,3,3 ; Ne: 10,10,10 (The number of neutrons is flexible. In fact, for Li 4 neutrons is more common than 3.)
b) Ne is heaviest because it has the most protons and neutrons.
c) Li is largest.
d) Mass is simple; size is subtle due to the wave-like nature of electrons and the critical role that electrons play in establishing atom size. This was not understood until the wave equation for electrons was discovered and solved around 1930.

4. H2 molecules go faster by a factor of 4. This is because they are 16 times less heavy (H2: amu=2 (2 protons); O2: amu=32 (16 protons; 16 neutrons)). To be at the same temperature means that their m v^2 must be the same, so to balance of the 16 times less mass one needs 4 times more speed (to get the same K.E.), since v is squared and m is not.

5. a) I'll upload a picture later. The lowest point is P.E. = 0 at x =0. At x=1 m, P.E. = (1/2) 8 (1^2) = 4 Joules

b) There is no friction, but as it moves its K.E. transforms into P.E. and it slows down. It starts with a K.E. of (1/2) m v^2= 16 J, and it will go until all its K.E. is transformed to P.E., which is x=2 m (where P.E. = (1/2) (8 J/m^2) (2^2 m^2)= 16 J).

c) It will slow down, stop and reverse direction at x=2. Then it will speed up to the left and go as far as x=-2, where its P.E. is also 16 J, and it will turn around there and go back-and-forth forever between -2 m and 2 m as long as there is no friction.

d) With a starting speed of 4 m/s, its motion is confined to the range -2 m to +2 m. This confinement comes about due to conservation of energy.

6) If k=2 J/m^2 instead of 8 J/m^2, it would be able to go twice as far, all the way to x=4 m, before loosing all its K.E. since the potential is 4x weaker (smaller). A potential with k=2 instead of 8 (J/m^2) is thus weaker and less confining.