What is the relationship between volume, spacing of energy levels, and entropy?

[weng cheong]

currently I'm studying the topic entropy. According to my text, entropy will increase when the accessible energy levels (microstates) increase. One of the approach to achieve this, we can decrease the spacing between energy levels by increasing the volume of the system.

i'm confused with this idea, i need a more detailed explanation on the change in spacing between energy level.

 

[Dale]

They are referring to the energy levels of a quantum mechanical particle in a box. It can be derived from first principles in quantum mechanics.

http://en.wikipedia.org/wiki/Particle_in_a_box#Energy_levels

 

[weng cheong]

i'm still a pre-U student, so if i would like to know the derivation of the equation for energy level, i have to study quantum mechanics?

 

[Dale]

Yes. At least, I am not aware of a non-quantum mechanical derivation.

 

[weng cheong]

thanks a lot =)

 

[mathfeel]

Just consider a particle in a box. Energy is $H=p^2/2$, but momentum is related to wavenumber, which is quantized by the boundary condition that the wavefunction vanishes at the edge. So allowed momentum is related to $L$ by $p_n=n\hbar/L$. So energy spacing is inversely proportional to $L^2$.