# Homework Help: Internal energy and specific heat

1. Feb 23, 2006

hello

First question-Internal energy W(I) of an ideal gas consists only of internal kinetic energy (thermal energy) W(KE).

W(I) = W(KE)

If at temperature T there are N molecules in an ideal gas,then total W(I) is

W(I) = N*3/2*k*T

We see from above formula that W(I) of ideal gas depends on number of molecules (N) and temperature T, but it doesn't depend on type of gas, that is it doesn't depend on what mass of gas molecules is.

M - relative molecule mass
u - 1.66*10^(-27)

But since N = m / (M*u)

W(I)=m*(3*k/2*M*u)*T

W(I)=m*c*T

c - specific heat - (3k/2*M*u)

Specific heat depends on size of relative molecule mass of gas. Thus the heavier the molecules, the smaller is specific heat.
Individually formulas for both specific heat and for W(I) make sense, but when we include c into formula for W(I), it gets a little confusing. Looking at original formula we see that W(I) doesn't depend on relative molecule mass, but if we look at W(I) = m*(3k/2*M*u)*T, we see that W(I) does depend on M.

second question - In ideal gas W(I) = W(KE)

N...number of molecules
T...temperature

W(i)=N*3/2*k*T

I've read that From this formula we can conclude that W(I) doesn't depend on pressure and volume. If at T=const. the ideal gas is shrinked,its internal energy won't change.

How would this formula give us such a clue?

Doesn't pressure increase when gas is compressed? Doesn't that happen because the resultant force appears in the matter that resists further deformation?
Doesn't that casuse change in internal energy? Don't we supply external energy to gas if we try to compress it?

Or doesn't at least internal energy(part of it) transform from kinetic to some other form?

thank you

2. Feb 23, 2006

### vaishakh

Question 2 - If we shrink the gas without changing the Tempreature, we are increasing pressure by decreasing the volume. Thus PV does not change, and so in nRT, and so the internal energy remains same.
You have already agreed that E = 3nkT/2. Here if we don't cahnge n, or the gas is put in a container or a closed sstem so that the number of moles of gas in our system does not change then k is Boltzmann's constant. Thus only variable is temperature nad we get that the internal energy is directly proportional to the temperature.

I didn't understand you question1

3. Feb 23, 2006

### Staff: Mentor

I think you are asking: "When you compress a gas, aren't you doing work on it? What happens to that energy?"

Yes, you are certainly doing work on the gas in compressing it. But realize that the only way you can compress a gas without increasing its temperature is by removing heat from it. The amount of heat energy removed must equal the work done so that the temperature remains constant.

If there were no heat flow from the gas, then the work done would lead to increased internal energy and thus higher temperature.

4. Feb 23, 2006

I'm sorry but I can't think of a better way to ask question 1. I hope you can figure it out though since I really would like to understand it!

But text said that W(i)=N*3/2*k*T for ideal gas doesn't depend on pressure and volume. If we have to remove amount of heat that equals the work done on gas, then that would suggest that W(I) does depend on pressure and volume!

That was going to be question for a new thread and I guess it is sort of related to this thread https://www.physicsforums.com/showthread.php?p=920051#post920051

Does PV mean Pressure * volume? I assume for nRT that n stands for number of molecules, T for temperature, but I have no idea what R stands for?

Perhaps this is not totally related to this thread, but why if n doesn't change is k Boltzmann's constant? What would k be if n did change?

5. Feb 23, 2006

### Staff: Mentor

That's true, it depends on T; it does not depend on pressure and volume (at least not directly). If you arrange for T to remain the same, then the internal energy doesn't change. Of course T is certainly related to P & V through the ideal gas law.

What makes you say that? W(I) still depends only on T. If you keep T constant, then W(I) remains constant. You stipulated:
Well, the only way you can do that--keep T constant as you compress it--is to arrange for heat to be extracted from the gas. If you enclose the gas in a thermally insulating container, then compress it, the temperature would increase--and thus, so would the internal energy. You're the one who said "keep T constant".

Yes, PV means Pressure X Volume. R is the universal gas constant.

k is Boltzmann's constant--period. It has nothing to do with n.

6. Feb 23, 2006

### Staff: Mentor

You are deriving the specific heat per unit mass (of a monatomic ideal gas). That's fine. But if you found the molar specific heat, you'd find that it's a constant for any monatomic ideal gas.

But m/(M*u) is just the number of molecules! Of course, for a given mass of gas the number of molecules depends on the mass per molecule. But the internal energy depends only on the number of molecules and the temperature, not on the mass. Meaning: Give me the same number of molecules of two different monatomic ideal gases at the same temperature and they will have the same W(I). (Of course, the two gas samples will have different masses.)

7. Feb 24, 2006

The more I read on thermal energy more confused I get

1-If we have different types of substance all containg same amount of atoms,then the amount of heat required to raise the temperature of each will depend solely on how much of heat does individual substance transform into other types of internal energy?
And mass of atoms has nothing to do with it?

2-With specific heat things are different since there we don't compare substances with the same amount of atoms but rather with same mass?

3-But why do we always requite same amount of specific heat to raise temperature of substance by one kelvin?

4-Don't conditions change depending on the current temperature of substance. Perhaps substance at lower temperature converts more heat into other types of energies than at higher temperatures?

5-Do 1000 atoms of steel at T=const. contain same amount of thermal energy as 1000 atoms of some gas at the same temperature?

But why would internal energy of monatomic gasses not depend on mass of molecules, but it does depend in non-ideal gasses (I'm assuming it does in liquids)?

What is molar specific heat?

I still have 30 pages of text to digest before I can start learning ideal gas law-bummer

8. Feb 27, 2006

### Staff: Mentor

Don't panic. This stuff can get complicated quick. And I am no expert on the detailed quantum models needed to accurately describe specific heat. (And you are probably not prepared for that digression anyway.)

It's not so much the mass of the atoms that counts, but the intermolecular forces (which provide a form of potential energy) and the degrees of freedom (the translational, vibrational and rotational modes of kinetic energy).

Right. If you use the molar specific heat (heat needed to change the temperature per mole of substance), then you'll note that different ideal gases (and different metals) will have pretty close to the same molar specific heat.
That's approximately true for many common subtances, but things get complicated quickly.

Things are different at different temperatures for many substances. Certainly the models used to explain specific heat must be modified for low temperatures.

No, the internal energy is different. But they have the same average translational kinetic energy.

What changes for non-ideal gases (and for liquids) are the intermolecular forces. In fact for some liquids (like water) most of the specific heat can be attributed to the energy needed to overcome those forces, not just increase the kinetic energy.

It's the specific heat per mole, not unit mass.

Not sure what text you are using, but many of your (good) questions can quickly lead into some complicated territory (physical chemistry, solid state physics). Learning the ideal gas law is much more fundamental that worrying about subtleties of specific heat.

A good source of info is hyperphysics: http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/spht.html#c1