# Energy levels for mass confined to 1D box

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1. Oct 8, 2014

### SalfordPhysics

1. The problem statement, all variables and given/known data
For a nitrogen molecule, calculate the lowest 2 energy levels and the characteristic temperature;
Mass of molecule = 2.33x10-26[kg]
Length of box = 10-9[m]

2. Relevant equations
E = n2.h2/8mL2 (n=1,2,3,...)

Characteristic Temperature (Tc) -> when thermal energy kBT = energy spacing of lowest 2 levels.

3. The attempt at a solution

E1 = h2/(8).(2.33x10-44)
= 2.358x10-24 [J]

E2 = n2.(E1) = 4(E1) = 9.432x10-42

Tc = (E2 - E1)/kB = 7.074x10-24 / kb = 0.512 [Kelvin]

The final answer quite simply does;t seem right to me, but I have checked numerous times my calculations. Can anyone clarify please.

2. Oct 8, 2014

### Staff: Mentor

Why does it sound wrong?

3. Oct 8, 2014

### SalfordPhysics

Just doesn't seem like 0.5 Kelvins is a reasonable answer. Main problem is that the term characteristic temperature is a new one and not quite sure of its understanding.

4. Oct 8, 2014

### nasu

Unless you have some common experience with nanometer size boxes, how can you even expect that you can tell what is a reasonable answer?
Your calculations seem OK. The result means that the system will be in the ground state only at very low temperatures. Otherwise will be very likely in excited states, due to the thermal excitation.

5. Oct 9, 2014

### SalfordPhysics

Thanks nasu, this makes sense.