[Luke149]

Hi, I'm stuck on a question about radial wave functions from a past paper. The question shows two radial wave functions for the hydrogen atom (shown Below).



The question then asks witch wave function corresponds to the higher energy state? and How you can tell this from just looking at the wave functions?

Since the wave is not periodic you cant infer its energy by looking at wavelength so the answer Ive come up with so far is that B shows the higher energy state because it has a higher average curvature and therefore it has a higher kinetic energy. Does this sound correct or have i missed something?

Many Thanks

Luke

[Wald_ron]

Larger wave function = higher energy
also witches ride brooms.

[Luke149]

Thanks for the reply, so then A represents the highest energy orbital because it is larger? Is the reason behind this the fact that the x-axis represents radius and a wave function with a greater radius would represent a larger orbital and therefore a higher energy orbital?  Also what does "witches ride brooms" refer to?

Many thanks

Luke

[FreeTheBee]

Larger wave function = higher energy

The energy is related to the curvature (2nd derivative).

Check here for some shapes of the radial distributions,
http://www.everyscience.com/Chemistry/Inorganic/Atomic_Structure/c.1101.php

[Luke149]

Thanks for the link was very helpful. From the examples in the link it seems that B represents a 2s orbital and A represents a 1s orbital therefore B represents the higher energy state. But im really confused as to why B represents the higher energy state, how do you tell if the wave function is larger because to me A looks larger than B?

Thank You

Luke

[tamim83]

But im really confused as to why B represents the higher energy state, how do you tell if the wave function is larger because to me A looks larger than B?

With plots of the radial function, it is really hard to tell so "size" won't help.  Since we know that plot A is for the 1s orbital and plot B is for the 2s orbital, we can say that the radial function with the larger number of nodes (places where the wavefunction has zero amplitude) will have a higher energy.  This is only when the l quantum number is the same, then the number of radial nodes only depends on n (there are n-l-1 radial nodes in a wave function). 

[Yggdrasil]

With electrons in an orbital, its energy is composed of two components: kinetic energy and potential energy.  As many have mentioned before, the kinetic energy of the electron is related to the curvature of the wavefunction and B clearly has a greater curvature and therefore more kinetic energy than A.

What about potential energy?  Here, the potential energy increases (becomes less negative) as the average position from the nucleus increases.  Here, A is on avreage farther from the nucleus than B, so A has more potential energy than B.

Which one has more overall energy?  Well, that depends on whether potential energy or kinetic energy dominates.  Clearly, the potential energy term must be greater than the  kinetic energy term or else the electron would not be bound by the atom and fly off into space.  Therefore because the potential energy term has more importance than the kinetic energy term and A has more potential energy than B, the electron with wavefunction A has a higher overall energy than an electron in wavefunction B.

[FreeTheBee]

Shouldn't you use the radial distribution function for this argument?

[Luke149]

Thanks for all the help when i read the post by tamim83 and how to work out which wave function represents the highest energy orbital buy the number of nodes since l is the same for the 1s and 2s orbitals and we already know that the 2s orbital is a higher energy orbital. However the post by Yggdrasil indicates that the answer is the other way round is this because Yggdrasil is referring to the radial distribution function which is related to the probability of finding an electron in a certain ares a Freethebee said. Thanks for everyones help

Luke