Finding the vibrational partition function of a diatomic molecule

In summary, the vibrational partition function is a thermodynamic quantity that describes the distribution of energy levels in a molecule at a given temperature. It takes into account the vibrational states of the molecule and can be calculated using the expression Qvib = ∑g(v)e<sup>-E<sub>vib</sub>/kT</sup>. This calculation is important in thermodynamics as it allows us to calculate properties such as internal energy and entropy, and it is directly proportional to temperature. However, there are simplifications often made in the calculation, such as assuming the molecule is in its ground state and neglecting anharmonic effects.
  • #1
thegirl
41
1
Hi,
How did they break down the following summation?
Screen Shot 2016-03-27 at 18.46.06.png

When finding the vibrational partition function ofa diatomic molecule it was approximated that the energy levels of the vibrational part of the diatomic molecule were harmonic and therefore the energy equation for a harmonic oscillator was used. Is the summation made to equal (1 - exp( - hw/KbT))^-1 because there are two degrees of vibrational freedom and therefore two energy levels? Or isit just due to maths?
 
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  • #2
It's just the math:
$$
\sum_{n=0}^{\infty} x^n = \frac{1}{1 -x}
$$
for ##|x|<1##.
 
  • Like
Likes vanhees71 and thegirl
  • #3
Omg, thank you!
 

Related to Finding the vibrational partition function of a diatomic molecule

1. What is the definition of vibrational partition function?

The vibrational partition function is a thermodynamic quantity that describes the distribution of energy levels in a molecule at a given temperature. It takes into account the vibrational states of the molecule, which are the different ways in which the atoms in the molecule can vibrate.

2. How is the vibrational partition function calculated for a diatomic molecule?

The vibrational partition function for a diatomic molecule can be calculated using the expression Qvib = ∑g(v)e-Evib/kT, where ∑g(v) is the sum of the degeneracy of each vibrational energy level, Evib is the energy of that level, k is the Boltzmann constant, and T is the temperature.

3. What is the significance of the vibrational partition function in thermodynamics?

The vibrational partition function is important in thermodynamics because it allows us to calculate thermodynamic properties such as the internal energy, entropy, and free energy of a molecule at a given temperature. It also provides insight into the vibrational motion of molecules and how it contributes to their overall energy.

4. How does the vibrational partition function change with temperature?

The vibrational partition function is directly proportional to the temperature. As the temperature increases, more vibrational energy levels become populated, leading to a higher value for the partition function. At very high temperatures, the vibrational partition function approaches the classical limit of 1.

5. Are there any simplifications or approximations made when calculating the vibrational partition function?

Yes, there are several simplifications that are often made in calculating the vibrational partition function. These include assuming that the molecule is in its ground electronic state, neglecting anharmonic effects, and using the harmonic oscillator approximation for the vibrational energy levels. These simplifications can lead to some inaccuracy in the final result, but they make the calculation more manageable.

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