Can Internal Vibrations Affect Temperature and Specific Heat?

[Infrasound]

In thinking about specific heat, I have been unable to resolve a particular issue.

If molecules in a substance are bumped and move faster, then they will measue to be at a temperature proportional to the speed they are moving which is a consequence of how hard on average they were hit(heat energy).

The law of conservation of energy suggests (to a very simple mind) that all substances of equal mass will require the same amount of energy to increase in temperature by a specific amount.

I know this is not true.

I also know that molecules can vibrate internally (the atoms in a molecule jiggle relative to each other). I think i have heard this referred to as degrees of freedom.

THE QUESTION: Why wouldn't the internal vibrations contribute to temperature? Wouldn't the vibrating atoms of a molecule in their oscillations bump out and hit other molecules and speed them up, thus making the temperature read at what would be expected to a lay person.

Is this really what causes varying specific heat capacities? If so, what is restricitng the internal vibrations from contributing to the temperature reading of the substance?

 

[mohammed.omar]

According to statistical mechanics, the vibration of molecules contribute by $$\frac{1}{2} k T$$ per molecule to the internal energy. Thus for solids, we have 3 Degrees of freedom for translational motion and 3 for the vibrational motion.

 

[Infrasound]

As indicated in my language above, I am a bit of a lay person on the topic. I still am unable to build a mental picture of what is happening.

I do appreciate the help, but you just gave me the names of things, rather than the explanation of what is going on.

The only thing I can come up with, would be an effect that would be like the opposite of a combustion reaction, where energy is used to move atoms (which are attracted in a certain configuration because of a bond) apart. Kind of like two magnets having enough energy to come apart, but the attraction resists/eats up some of that energy (stored as potential until/if they ever come back together with sufficient force).

This explanation seems to agree with the law of conservation of energy. Does it agree with the rest of our observations of the behavior of materials?

Anybody know?

 

[mohammed.omar]

Let me put it in simple language.

Internal vibrations do contribute to the internal energy. The particles contribute to the internal energy in the form of kinetic energy, during their motion around their equilibrium position, and contribute in the form of potential energy, during their vibrations around their equilibrium position in the lattice.

So simply put: Yes. The internal energy is affected by the vibrations of the atoms in a solid.

Concerning the specific heat variation. At room temperature, all the solids have a molar specific heat equal to 3R.i.e. it is constant. This is called the Dulong-Petit law. That is to say, 1 mole of any solid would require the same amount of energy to raise its temperature, which agrees to what you are trying to say.

To sum up, your arguments are correct in terms of the contribution of the atoms vibration in the specific heat, and that the molar specific heat should be constant for all solids.

The variation of the specific heat, not the molar specific heat, can be attributed to the variation of the number of atoms in one unit mass from one material to another, causing a change in the specific heat. This is not the case for molar specific heats as the number of atomes in 1 mole of a substance is constant.

Let me know if this is better than the first post :).. Let me know if you need any further explanation.

Regards

 

[Infrasound]

-Thank you for the simpler language. I am not very bright and seem to panic when it comes to complicated language in explanations. I appreciate your patience.

-Wow, I didn't even think about the Dulong-Petit law. Let me make sure I have it straight... A consequence of the Dulong-Petit law would be that two substances of unequal mass (but same number of atoms/molecules) COULD have the same specific heat based solely on the fact that there are an equal number of molecules in the two substances.

I didn't even really think of that consequence. For some reason, my brain doesn't like it. It make sense though. This seems to really expose the fact that temperature is not just the speed of the particles but also the mass.

 

[mohammed.omar]

This conversation is really nice Infrasound, your point of view is very intuition driven and is helping me a lot think about the problem. Thanks to you :).

If we neglect some non-idealities and differences between materials and neglect the quantization effects that occur at low temperature, then Dulong-Petit law that the molar specific heat of two solids IS constant and the same for all metrials.

Thinking about the equality in the molar specific heat because there is the same number of atoms/molecules in one mole is quite satisfying as these two materials would have the same amount of internal energy at a given temperature.

Thinking of the mass as something that would reflect on temperature does not seems right to me. I would better look at it as increasing the mass would increase the amount of internal energy stored, because it would mean containing more atoms/molecules.
This is reflected as a change in the heat capacity and the amount of energy needed to raise the temperature of a body of a given mass, but then it is because the number of particles to which you distribute the added temperature has increased, so you actually give each indvidual atom/molecule a diminished amount of energy and hence you will need a larger amount of energy to increase the temperature of a larger mass (with more number of molecules).

 

[Infrasound]

This conversation is really nice Infrasound, your point of view is very intuition driven and is helping me a lot think about the problem. Thanks to you :).
.

No problem at all. I'm glad you mention this.

I have found that when looking at things with mental pictures, I tend to maybe notice things about nature that I would probably never otherwise notice.

It seems to me that many people prefer to just learn the formulas and then calculate, but there is a bit more to it than that. It's more fun for me.

 

[Infrasound]

Thinking of the mass as something that would reflect on temperature does not seems right to me. I would better look at it as increasing the mass would increase the amount of internal energy stored, because it would mean containing more atoms/molecules.
This is reflected as a change in the heat capacity and the amount of energy needed to raise the temperature of a body of a given mass, but then it is because the number of particles to which you distribute the added temperature has increased, so you actually give each indvidual atom/molecule a diminished amount of energy and hence you will need a larger amount of energy to increase the temperature of a larger mass (with more number of molecules).

This was exactly the problem that I had. For some reason, I was insisting that more mass meant more heat to reach a particular temperature, but that is only true when the increase in mass is the result of an increase in particles.

 

[theprofessor0]

I HAV A Q TO MOHAMMAD:
its abt the dulong and petits law. i know that it says that the sp heat and the wt of higher elements multiply to give 6 cal..., and the sp heat (molar) OF the substances is equal to 3R at RTP. but i dnt think i believe this. i want to know one thing, the total no of molecules need the same heat: accepted. but molecule of one substance may be heavier than the other, and thus may need more energy to vibrate. so, this law appears lil bit OUT OF ORDER, doesn't it?

 

[mohammed.omar]

Hello theprofessor0,

Sorry for the late reply.

Your intuition about the mass of the particle is very beautiful but it omits the fact that the particles of the same energy but different masses do not need to oscillate with the same frequency or the same amplitude, they might have a lower amplitude and/or frequency for higher mass oscillators.

To formalize what I'm saying with some mathematics let's consider the potential energy of a given degree of freedom, say motion in the x-direction; in such a case the average energy (1/2 kT) can be considered to be equal to the average potential energy (1/2 K A²) where the symbols are as follows:
k: Boltzmann constant
T: Absolute temperature
K: Equivalent elastic constant between the molecules
A: Amplitude of the oscillations

For simple oscillatory motion we know that $$\textrm{K} = \omega^2 \textrm{m}$$

That is we can say:
$$\frac{1}{2} k T = \frac{1}{2} K A^2 = \frac{1}{2} \omega^2 m A^2$$

For particles with heavier masses, the oscillation frequency/amplitude will be lower to achieve the equality with the average energy associated with them.

The same occurs for the Kinetic energy, where you'll find that heavier atoms will have lower speeds for the same temperature.

Let me know what you think about this. I really like the intuitive way you are approaching this problem.

 

[theprofessor0]

yes. thank you. i get it