What is the quantum of energy for an atomic oscillator in a block of magnesium?

[kkranz_gatech]

The interatomic spring stiffness for magnesium is determined from Young's modulus measurements to be 12 N/m. The mass of one mole of magnesium is 0.012 kg. If we model a block of magnesium as a collection of atomic "oscillators" (masses on springs), what is one quantum of energy for one of these atomic oscillators? Note that since each oscillator is attached to two "springs", and each "spring" is half the length of the interatomic bond, the effective interatomic spring stiffness for one of these oscillators is 4 times the calculated value given above.

Use these precise values for the constants:
hbar= 1.054610e-34 J · s (Planck's constant divided by 2)
Avogadro's number = 6.022110e23 molecules/mole
k = 1.380710e-23 J/K (the Boltzmann constant)

I tried that one quantum of energy would be hbar*sqrt(ks/m) = 1.054e-34*sqrt((4*12)/(.012/6.02e23)) which is incorrect. The 'hint' is that one quantum of energy is the amount of energy required to raise one atomic oscillator from one energy level to the next highest energy level

 

[gtzpanditbhai]

That equation is exactly right, but make sure you are putting in at least 5 numbers for the answer just like the question asks.

 

[jchojnac]

whenever I calculate the energy, I get an error on my calculator b/c it can't do 4.91e26 factorial. How did you input that as an answer?

 

[yoyoz41]

I can't get the right answer with that equation, too

hbar*sqrt(ks/m) = 1.054e-34*sqrt((4*12)/(.012/6.02e23) = energy

btw, Jchohnac, where does the "factorial" come from?

 

[jchojnac]

i got the first question right with that equation, but I'm talking about the table and calculating the energy.

 

[gtzpanditbhai]

For the energy all you need to do is multiply the energy of one quantum by the number of quantums.

So if 5.17597e-21 was the energy of one quantum, multipl that by 20 and you get the enegery. For entropy, it is just k*ln(# of ways) given in the table

 

[yoyoz41]

hey, Jchohnac, how did u calculate out the energy of one quantum ??
is it hbar*sqrt(ks/m)?
or hbar*sqrt(4*ks/m)?

 

[jchojnac]

1.054e-34*sqrt((4*12)/(.012/6.02e23))...i followed this with my numbers

 

[jchojnac]

Does anyone know how to do the very last part on question 1?...

There are 100 atoms in this object. What is the heat capacity on a per-atom basis? (Note that at high temperatures the heat capacity on a per-atom basis approaches the classical limit of 3k = 4.210-23 J/K/atom.)

Heat capacity per atom = ?? J/K/atom

 

[jchojnac]

never mind, i got it

 

[nitefalls]

I still dun get the Heat Capacity per atom.. Can anyone help me? Thanks

 

[AirChambz]

One Quantum/ delta Temp/ 100 atoms = Heat Capacity