Temperature as a function of vibration energy quanta

In summary, the conversation focused on a small cluster of four copper atoms and their independent oscillation around equilibrium positions. Each direction of vibration was modeled as a harmonic oscillator with quantized energy. The number of microstates and entropy were already calculated, and the discussion then moved on to finding the function for temperature and heat capacity per atom. The suggested approach was to use the equation T = dE(int)/dS, assuming that n is much larger than the number of oscillators.
  • #1
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Homework Statement


Consider a small cluster of four copper atoms. Assume, that each atom can oscillate around its equilibrium position independently of the other atoms. Let us model each direction of vibration as a harmonic oscillator, whose energy is quantized
Evib = ¯hω(n +1/2), where ω = sqrt(k/m), k is the ’spring constant’ of the bond between the atoms (k ≈ 120 N/m) and m is the mass of the copper atom. Compute as the function of vibration energy quanta n = 0,1,2,3,... the (a) number of microstates Ω
(b) entropy S
(c) temperature T
(d) heat capacity C / atom

Homework Equations

The Attempt at a Solution


I have already functions for the number of microstates and the entropy S:
Ω(n) = ((n+11)!) : (n!11!)
S(n) = k*ln (((n+11)!) : (n!11!)
Now I try to find the function T(n) and started with:

1/T = dS/dEint (dS = change in entropy, dEint = change in internal energy)

Can I put this equations as T = dEint/dS ?
 
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  • #2
Yes you can do that, but you'll have to put E(int) in terms of S. It may help if you're allowed to make the assumption that n >> # of oscillators.
 
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Related to Temperature as a function of vibration energy quanta

1. What is the relationship between temperature and vibration energy quanta?

The relationship between temperature and vibration energy quanta can be described by the Boltzmann distribution, which states that at a given temperature, the energy of a molecule is distributed among its vibrational states. As temperature increases, the average energy per molecule also increases, resulting in more frequent and higher energy vibrations.

2. How does the temperature of a substance affect its vibrational energy?

The temperature of a substance directly affects its vibrational energy, as higher temperatures result in higher average kinetic energy of molecules, leading to stronger and more frequent vibrations. Conversely, lower temperatures result in lower vibrational energy.

3. Can the temperature of a substance be determined by its vibrational energy?

Yes, the temperature of a substance can be determined by measuring its vibrational energy. This is often done using techniques such as infrared spectroscopy, which can measure the frequencies and intensities of vibrational energy transitions and provide information about the temperature of the substance.

4. How do different substances respond to changes in temperature and vibrational energy?

Different substances may respond differently to changes in temperature and vibrational energy. For example, substances with stronger intermolecular forces, such as solids, may require more energy to increase their temperature and vibrational energy. Additionally, different substances may have different vibrational energy levels and modes, resulting in unique responses to changes in temperature.

5. Is there a limit to how high or low the temperature and vibrational energy of a substance can be?

There are theoretical limits to both the highest and lowest possible temperatures and vibrational energies of a substance. The highest possible temperature is known as the absolute hot and is believed to be around 1.416 x 10^32 Kelvin. The lowest possible temperature, known as absolute zero, is the point at which all molecular motion stops and is equal to 0 Kelvin or -273.15 degrees Celsius.

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