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In: Physics

Thermal energy ~ kT , where T is the absolute temperature (Kelvin) and k is Boltzmann's...

Thermal energy ~ kT , where T is the absolute temperature (Kelvin) and k is Boltzmann's constant is an important benchmark... it is the energy available to electrons from their surroundings by virtue of random thermal motion.

The value of kT at room temperature is 0.026 eV or 26 meV.

What is the size, L, of the 3-D "box" in nanometers (a cube of side L) such that the energy difference between the ground state (1,1,1) and the first excited state (1,1,2) is about kT at room temperature?

Expert Solution

The energy of the n th state in the 3D box is given by

So, the energy of the ground state (1,1,1) is given by


And the energy of the 1st excited state (1,1,2) is given by


So, the difference between these two energies is given by


  And if this of the order of kT, then, we get



And so, putting the given values,

we get the size L as

genius_generous answered 2 months ago

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