# [thermodynamics] simple question + view of temperature?

1. Dec 28, 2008

### nonequilibrium

Hello. I'm trying to thoroughly grasp Thermodynamics, so I'm looking into it again, starting from scratch.

1) Quite early, I had a question with something very commonly taught. If you double the volume of a gas (the temperature is taken as a constant), the pressure becomes half its original value. Pressure is defined as F/A, and as T = constant => v = constant => no change in momentum => no larger/smaller force, so the only variable that changes in the defintion of pressure is the area A. However, to get the new pressure, the area has to be twice as large if you fill it in, in the defintion. But of course, if you double the volume, this doesn't mean you double the area. So... How come pressure really does become half as large?

2) I find it quite confusing how ideal gasses and realistic ones are used through each other. I'm uncertain about the definition of temperature. So temperature only is proportional to the internal energy with ideal gasses? Temperature only says something about the moving of the particles, right? And the internal energy also says something about the intermolecular forces. Does temperature also include vibration and rotation? How come there are different specific heats? Is this due to the presence of realistic gasses? Would it be a constant for ideal gasses? And are they different because the intermolecular forces are different + different masses?

Sorry for unloading a lot of question rapidly after each other, but I'm having trouble truly understanding it all. I'm learning myself thermodynamics using the hyperphysics site, should that work? Other helpful sites are always welcome...

2. Dec 28, 2008

### stewartcs

That is the wrong equation. If you are assuming an ideal gas then the equation of state would be PV = nRT. Since the temp is held constant and the quantity of gas is the same (R is always constant) then by doubling the volume the pressure would have to be half in order for the RHS of the EOS to remain constant.

The internal energy of an ideal gas is directly proportional to the temperature. The specific heat is a function of temperature as well but can sometimes be assumed constant. The specific heats vary depending on the type of gas since they behave differently in nature. The specific heat is also dependant on the degrees of freedom. For a diatomic molecule like nitrogen the number of degrees of freedom are typically 5 (3 translational and 2 rotational). However, at high temperatures the molecule will begin to vibrate and 2 more DOF are added). Take a look at the first attachment to see a typical model of the molar specific heats as a function of DOF and temperature. The second attachment shows the specific heat (constant pressure) of various gases.

What book are you using for your study of Thermodynamics? More specific information will be in it.

CS

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3. Dec 28, 2008

### stewartcs

I recommend using a text book if you are not already. Alternatively, MIT has a site that will probably be more beneficial for you:

http://ocw.mit.edu/OcwWeb/web/courses/courses/index.htm

CS

4. Dec 29, 2008

### Andrew Mason

If T is constant, then the energy of the molecules of the gas is constant. Pressure is F/A or $F\cdot d/V$. The numerator has units of energy. So pressure is a measure of energy density. As V increases, P must decrease proportionately in order to maintain constant energy (T is constant).

A better way to look at it is at the molecular level. The force exerted on the wall of the container by a molecule in an elastic collisiion is:

$$F = \Delta P/\Delta t = 2mv/\Delta t$$

where $\Delta t$ is the duration of the collision. With large numbers of molecules continuously making small collisions, the duration of the collision is effectively the average time between collisions.

A molecule traveling at an average speed v in a container whose walls are distance L apart will have two such collisions in time $\Delta t = 2L/v$. This is essentially the average time that it takes for a molecule to bounce off a wall, travel the length of container bounce off the other wall and return to the original wall to begin another collision with the first wall. So the force on the first wall, averaged over time, is:

$$F = 2mv/2L/v = mv^2/L$$

$$P = F/A = 2mv/(2L/v)L^2 = mv^2/L^3 = mv^2/V$$

As volume increases with average molecular speed remaining constant, P must decrease in inverse proportion to volume.

You should review the kinetic theory of gases. Temperature is a measure of the translational energy of gas molecules. It does not include vibration and rotation. The specific heats of different gases occur in large part due to different molecular moments of inertia (giving different degrees of freedom such as rotation, vibration as well as translation) and because of the principle of equipartition of energy (meaning energy is distributed equally between the degrees of freedom).

For a monatomic gas with only translational freedom, energy is all translational kinetic energy so any addition of heat adds to translational energy and, therefore, temperature. For a diatomic gas with rotational and translational freedom, some of the heat flow is absorbed by the molecules as rotational energy which does not add to temperature. So it takes more heat to increase temperature. For a polyatomic gas with vibrational, rotational and translational freedom, the heat absorption is even greater so the ratio of heat flow to temperature increase is even lower.

AM

5. Dec 29, 2008

### nonequilibrium

CS - Yes, I got it from the universal gas law, but I was wondering how that would apply to the defining formula of pressure. Thanks for the attachments and the courses link. "The internal energy of an ideal gas is directly proportional to the temperature." not in every case though, right? As temperature doesn't pay attention to vibration & rotation (*points to Andrew Mason's post*), while internal energy does.

Mason - Oh how I love to see that derivation of that principle using p = F/A, exactly what I was looking for!
And about temperature... So it only measures the kinetic energy, not the vibration and rotation, which is categorized as potential energy of the atom, along with intermolecular forces. And different gases have different specific heats cause they have different potential energies even at the same temperature? And in a way, you could see the specific heat as a correction factor? Cause if you're supplying heat to a gas, hoping to raise its temperature, the specific heat "tells" you that a part of the heat will actually also be used to increase potential energy (due to the equipartition, as you say) as it increases temperature. (well actually, re-reading your last paragraph, that's exactly what you said, great

I also often see the term "kinetic temperature", I take it that is what we are talking about. Is there also another sort of temperature then? Which might take in account the potential energy and thus embody internal energy? Is there more than one "temperature" or is the word "kinetic" nothing but an epitheton ornans in this case?

Thanks for the help, by the way ;) much appreciated

6. Dec 29, 2008

### Andrew Mason

Different gases can have different specific heats because
a) their constitutent molecules have different degrees of freedom;
b) there are inter-molecular forces between the molecules of the gas and these forces differ between the gases;​

But if the gases have the same kind of molecules (eg both monatomic or both diatomic) and if the intermolecular forces are not significant and the volume of the molecules themselves is much less than the volume of the gas) their specific heats will be the same: $C_v = NR/2$ where N is the number of degrees of freedom.

I am not sure what you mean by correction factor. Specific heat is just the ratio of heat flow to temperature increase per unit of matter. If there are intermolecular forces, overcoming those forces takes energy so the heat flow will not all go to increasing the kinetic energy of the molecules. This is a kind of potential energy, since that energy can be recovered when the temperature is lowered.
The stronger these forces, the more heat required to raise the temperature, so the greater the specific heat.
There are various, mutually consistent ways to define temperature but the kinetic theory provides the generally accepted definition. Temperature is just translational kinetic energy of the molecules. All the intermolecular forces do is require more heat to be added to get the gas up to a certain temperature.

AM

7. Dec 29, 2008

### stewartcs

Yes, the internal energy of an ideal gas is always directly proportional to the temperature.

$$\Delta u = u_2 - u_1 = C_{v,ave}(T_2 - T_1)$$

The specific heat does depend on the DOF as I stated earlier so the internal energy does depend on the DOF.

CS

8. Dec 31, 2008

### nonequilibrium

Ah okay :) Thank you both for the clear explanations, it really helped me!