# Distance between hydrogen molecules in H2 using specific heat capacity

• enjoi668
In summary, the heat capacity of hydrogen gas increases with temperature due to the contributions from rotational energy states. At high temperatures, the molecules can be in their lowest rotational quantum state. This gives them an extra energy and the distance between the hydrogen atoms is estimated to be around 80k.
enjoi668

## Homework Statement

Below about 80 K the heat capacity at constant volume for hydrogen gas (H2) is 3/2k per molecule, but at higher temperatures the heat capacity increases to 5/2k per molecule due to contributions from rotational energy states. Use these observations to estimate the distance between the hydrogen nuclei in an H2 molecule.

## Homework Equations

boltzmann distribution (possibly)
C = deltaEatom / deltaT C=specific heat capacity

## The Attempt at a Solution

I honestly do not even have an idea on where to start this problem, any help at all, or hints on where to start, or some guidance on the equations I should use will be more than appreciated. Thanks

Here's a hint: the rotational energy states "freeze out" at low temperatures because individual H atoms in H2 lose their identity and become, very loosely speaking, a single particle. Classically, that's never going to happen: two balls connected with a stick isn't going to look any less like two balls and a stick at low temperatures. So think quantum mechanically. At what distance, roughly, would the hydrogen atoms lose their identities?

When the temperature is high enough, the rotation gives 2 more degrees of freedom, so the average energy of the molecules increases by kT. This term arises from the rotational energy, which is E=0.5 Iw^2 (I is the moment of inertia and w is the angular speed of rotation). The angular momentum of rotation is L=Iw, and it can be an integer multiple of h/(2pi) according to Bohr's quantum condition. The molecules can be in their lowest excited rotational quantum state, so L=h/(2pi). The average energy for the rotation is 80k. The moment of inertia of the hydrogen molecule is 2mr^2. From all of this, you can estimate the distance between the hydrogen atoms.

ehild

thank you very much

## 1. What is the specific heat capacity of hydrogen molecules in H2?

The specific heat capacity of hydrogen molecules in H2 is approximately 14.3 J/g °C at standard temperature and pressure.

## 2. How is the distance between hydrogen molecules in H2 related to its specific heat capacity?

The distance between hydrogen molecules in H2 is directly related to its specific heat capacity. As the distance between molecules increases, the specific heat capacity decreases, and vice versa.

## 3. What is the formula for calculating the distance between hydrogen molecules in H2 using specific heat capacity?

The formula for calculating the distance between hydrogen molecules in H2 using specific heat capacity is given by: d = 1/2 (N/A)^1/3, where d is the distance between molecules, N is the Avogadro's number, and A is the specific heat capacity.

## 4. How does the distance between hydrogen molecules in H2 affect its physical properties?

The distance between hydrogen molecules in H2 plays a crucial role in determining its physical properties. A larger distance between molecules results in weaker intermolecular forces, leading to lower boiling and melting points, as well as a lower density of the substance.

## 5. Can the distance between hydrogen molecules in H2 be manipulated?

Yes, the distance between hydrogen molecules in H2 can be manipulated through changes in temperature, pressure, and the addition of other elements. For example, increasing pressure can decrease the distance between molecules, while adding another element can increase the distance between molecules.

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