# Inter-atomic force in a Hydrogen molecule - finding spring stiffness

In summary, the stiffness of the "spring" representing the interatomic force between two H atoms in a Hydrogen gas molecule can be estimated using the observed increase in heat capacity at 2000K due to contributions from vibrational energy states. This can be done by considering the quantum harmonic oscillator approximation and using equipartition to calculate the energy state at 2000K, which can then be set equal to (1/2)*h-bar*omega to find the numerical value of omega and thus the stiffness of the interatomic bond.
Inter-atomic force in a Hydrogen molecule -- finding "spring" stiffness

## Homework Statement

At about 2000 K the heat capacity (at constant volume =) increases to (7/2)k per molecule due to contributions from vibrational energy states. Use these observations to estimate the stiffness of the "spring" that approximately represents the inter-atomic force between the two H atoms in a Hydrogen gas molecule (H2).

## Homework Equations

K=.5Iω^2
Ι=2mr^2
E=mCΔT
Y=(ks,i)/d

These are possibilities...not sure whether they are all the right equations to use...

## The Attempt at a Solution

Stress of interatomic bond = (ks,i)(s)/(d^2) where s is the stretch of the interatomic spring and d is the "original distance", which in this case would be the length of the interatomic bond between the two Hydrogen atoms in the H2 molecule (or the distance between the hydrogen nuclei in H2). I calculated (hopefully correctly) d already, and would only need stress and stretch to solve this equation. However, I have no idea if it even makes sense to be using this relationship (with stress and stretch) in this problem, one, because it doesn't make a lot of sense, and two, because it is from a much earlier chapter that we are not going through right now in class.

If you could help, I would greatly appreciate it!

Thanks.

A good approximation to the hydrogen molecule is a quantum harmonic oscillator. The ground state potential energy of the bond would then be (1/2)*h-bar*omega, where omega is sqrt(k/m). Since this vibrational energy state only starts contributing to the total energy at 2000K, you can figure out the numerical value of the energy state using equipartition and set it equal to (1/2)*h-bar*omega.

Oh, thank you! That helps a lot.

## 1. What is inter-atomic force?

Inter-atomic force is the force that holds atoms together in a molecule. It is the attractive force between the positively charged nucleus of one atom and the negatively charged electrons of another atom.

## 2. How is inter-atomic force measured?

Inter-atomic force is measured using various techniques such as atomic force microscopy, x-ray diffraction, and spectroscopy. These methods involve measuring the distance between atoms and the strength of the force between them.

## 3. What is a Hydrogen molecule?

A Hydrogen molecule is a molecule composed of two hydrogen atoms bonded together by a covalent bond. It is the simplest and most abundant molecule in the universe.

## 4. How is spring stiffness related to inter-atomic force in a Hydrogen molecule?

Spring stiffness is a measure of the resistance of a spring to deformation under a force. In a Hydrogen molecule, the inter-atomic force acts like a spring, and the spring stiffness is related to the strength of this force.

## 5. How can the spring stiffness in a Hydrogen molecule be calculated?

The spring stiffness in a Hydrogen molecule can be calculated using Hooke's Law, which states that the force applied to a spring is directly proportional to the displacement of the spring. By measuring the displacement and force between the two hydrogen atoms, the spring stiffness can be determined.

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