# Homework Help: Physics: Specific heat capacity of a solid

1. Dec 4, 2008

### mag_icando

1. The problem statement, all variables and given/known data
The diagram below shows a one-dimensional row of 5 microscopic objects each of mass 410-26 kg, connected by forces that can be modeled by springs of stiffness 26 N/m (so each object can be modeled as if it were connected to a single spring of effective stiffness 4ks = 104 N/m -- neglect any possible differences for objects near the ends). These 5 objects can move only along the x axis.

Use these precise values for the constants:
hbar = 1.054610-34 J ยท s (Planck's constant divided by 2)
k = 1.380710-23 J/K (the Boltzmann constant)

What is one quantum of energy for one of these objects?
Delta E = 5.37743e-21 correct check mark J

Using the Einstein model, calculate the entropy of this system for total energy of 0, 1, 2, 3, 4, and 5 quanta.
q = 0: S = 0 correct check mark J/K

q = 1: S = 2.22215e-23 correct check mark J/K

q = 2: S = 3.73900e-23 correct check mark J/K

q = 3: S = 4.90887e-23 correct check mark J/K

q = 4: S = 5.86590e-23 correct check mark J/K

q = 5: S = 6.67745e-23 correct check mark J/K

Calculate to the nearest degree the average absolute temperature of the system when the total energy is in the range from 3 to 4 quanta. (You can think of this as the temperature when there would be 3.5 quanta of energy in the system, if that were possible.)
T3 to 4 = 561.88730 correct check mark K

Calculate to the nearest degree the average absolute temperature of the system when the total energy is in the range from 4 to 5 quanta. (You can think of this as the temperature when there would be 4.5 quanta of energy in the system, if that were possible.)
T4 to 5 = 662.61229 correct check mark K

Calculate the heat capacity per object when the total energy is 4 quanta. (Think of this in terms of increasing from 3.5 quanta of energy in the system to 4.5 quanta of energy in the system, if that were possible.)
C4 = 1.067745e-23 correct check mark J/K/object

If the temperature were raised very high, classically what would we expect the heat capacity per object to be for this one-dimensional system? Give a numerical value.
Chigh T = ??? wrong check mark J/K/object
(One reason for the discrepancy is that the high-temperature limit assumes that the number of oscillators is large (N >> 1), which is not the case in this tiny system.)

2. Relevant equations

C= deltaE/deltaT

classical limit of 3k = 4.210-23 J/K/atom

3. The attempt at a solution

I got all the questions except the last one which I've already tried searching for clues in the internet.
I tried 3kN= 3*1.380710e23*5 but got it wrong.
I thought it was just supposed to be 3k like in the relevant equation but that's wrong maybe because N=5??
I also thought that 3 in the 3k equation has something to do with 3 springs, so I tried 2*k instead of 3*k but got it wrong too.

2. Dec 4, 2008

### phys2211

I don't think you need to multiply it by 5 if you're already multiplying by 3. Since you do 3k for a three-dimensional system, i figured that you could just do 2k, since only two springs are attached to each 'atom'. This was also wrong. I also tried 5k, but that didn't work. Apparently 3k is also wrong.

Now that we've eliminated 5 options, anyone else have ideas?

3. Dec 4, 2008

### mag_icando

I didn't know that 3k was for a three-dimensional system. If that's the case, then how about trying 1k since this is a one-dimensional system.

Thanks for joining the discussion, phys2211. And Thanks a lot for that important point you gave out.

4. Dec 5, 2008

### phys2211

glad i could be of service :)

5. Dec 2, 2009

### nitefalls

Hey how do u search for T3 to 4 on T4 to 5?
Thanks