Why Is My Calculated Debye Temperature for Sodium Higher Than Literature Values?

In summary: It's a bit complicated, but it might be worth looking into.In summary, the conversation discusses the calculation of the Debye temperature for sodium, which has a bcc structure with a molecular mass of 22.99 gr/mol and a mass density of 0.971 gr/cm^3. The average speed of sound in Na at room temperature (300K) is 3200 m/s. The attempt at a solution involves plugging in the numbers into an equation and obtaining a Debye temperature of 280.3K. However, this differs from the literature value of approximately 150K. Further discussion suggests that the discrepancy may be due to not accounting for the fact that sodium has two atoms per cell in its bcc
  • #1
MMS
148
4

Homework Statement


Na has a bcc structure with molecular mass of 22.99 gr/mol, mass density of 0.971 gr/cm^3.
The average speed of sound in Na (at room temperature=300K) is 3200 m/s.
Calculate the Debye temperature for Na

Homework Equations


I worked out this equation to calculate the Debye temperature (If needed I can show how)
XzxlsHT.png

The Attempt at a Solution


I plugged in all the numbers in the above equation with the correct units and I get that the Debye temperature is 280.3K. However, in literature I found that it is approximately 150K.
Is there something wrong with my calculations? Am I missing out on something here?

Thank you!
 
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  • #2
MMS said:

Homework Statement


Na has a bcc structure with molecular mass of 22.99 gr/mol, mass density of 0.971 gr/cm^3.
The average speed of sound in Na (at room temperature=300K) is 3200 m/s.
Calculate the Debye temperature for Na

Homework Equations


I worked out this equation to calculate the Debye temperature (If needed I can show how)
XzxlsHT.png

The Attempt at a Solution


I plugged in all the numbers in the above equation with the correct units and I get that the Debye temperature is 280.3K. However, in literature I found that it is approximately 150K.
Is there something wrong with my calculations? Am I missing out on something here?

Thank you!
Please provide your calculations.
 
  • #3
SteamKing said:
Please provide your calculations.
BnOvrQ7.png
 
  • #4
MMS said:
BnOvrQ7.png
No, I meant your numbers.
 
  • #5
SteamKing said:
No, I meant your numbers.
ShjbCQo.png
 
  • #6
Anyone?
 
  • #7
MMS said:
Anyone?
Your calculations are arithmetically correct, as far as I can see.

It's not clear that NA ⋅ ρ / MW is an accurate substitute for the N / V which is used in other Debye Temperature derivations I have seen. Your expression doesn't seem to account for the fact that since sodium is bcc, there are two atoms per cell, rather than one. It makes a difference in calculating the edge length of the cell.

I think for calculating a reasonable approximation to the Debye Temp., this is the issue which must be resolved. In some articles I have seen, the authors try to use the properties of the material measured close to the Debye Temp., like the speed of sound and the density, to come up with a more accurate calculation.
 

1. What is the Debye temperature for Na?

The Debye temperature for Na (sodium) is approximately 97.72 K (kelvin) or -175.43 °C (degrees Celsius). This value represents the temperature at which the majority of thermal energy in a solid is in the form of phonons (quantized lattice vibrations).

2. How is the Debye temperature for Na calculated?

The Debye temperature for Na can be calculated using the formula θD = (hbar/kB)(6π2n/V)1/3, where hbar is the reduced Planck's constant, kB is the Boltzmann constant, n is the number density of atoms in the solid, and V is the molar volume of the solid.

3. Why is the Debye temperature important for studying materials?

The Debye temperature provides important information about the thermal and mechanical properties of materials. It can be used to determine the heat capacity, thermal conductivity, and elastic properties of a solid. It also helps to understand the behavior of materials at low temperatures, such as in superconducting or magnetic materials.

4. How does the Debye temperature for Na compare to other elements?

The Debye temperature varies significantly among different elements and is influenced by factors such as atomic mass, lattice structure, and bonding strength. Generally, lighter elements have higher Debye temperatures, while heavier elements have lower Debye temperatures. For example, the Debye temperature for Na is higher than that of aluminum (92.1 K) but lower than that of iron (280.2 K).

5. Can the Debye temperature of a material change?

Yes, the Debye temperature of a material can change with temperature and pressure. As the temperature increases, the Debye temperature also increases due to the increased thermal energy and vibrations in the lattice. Similarly, applying pressure can also affect the Debye temperature by altering the atomic spacing in the solid.

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