Energy levels for mass confined to 1D box

[SalfordPhysics]

Homework Statement

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]

Homework Equations

E = n².h²/8mL² (n=1,2,3,...)

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

The Attempt at a Solution

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

E₂ = n².(E₁) = 4(E₁) = 9.432x10^-42

T_c = (E₂ - E₁)/k_B = 7.074x10^-24 / k_b = 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.

 

[mfb]

Why does it sound wrong?

 

[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.

 

[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.

 

[SalfordPhysics]

Thanks nasu, this makes sense.