# Is 1 degree K/C equivalent at all temperatures?

• Urshilikai
Even in the limit of ideal gas behavior, the heat capacity is a function of...The heat capacity of a real substance is not constant. It is a function of temperature. An ideal gas -- that's an ideal. Some gases do come close to this ideal over some finite temperature range, but there is no such thing as an ideal gas. Nor an ideal solid.

#### Urshilikai

This is an odd question. I was talking with a social scientist the other day and he seemed to think that the amount of energy it takes to raise the temperature 1 degree Celsius or Kelvin was minutely different depending on the temperature (ie near absolute zero the energy it takes to increase T by 1 is different than the energy it takes to increase T by 1 at 300K). I tried explaining to him that kinetic energy is linearly proportional to temperature, but he insisted that he was right and that the scale was very slightly nonlinear (no proof or source I'm afraid, so it's been driving me nuts :/ ). I have done some searching to no avail, I turn to you for ideas. Are there any nonlinear effects that cause the energy separation between degrees K/C to be unequal?

Maybe in his mind, he wasn't thinking of an isolated system, but a system on Earth, in which you're generally fighting strong gradients to keep any particular region cooled, and the colder you get, the bigger a gradient you generate, and the more the heat wants to get into your cooled region.

It just occurred to me that near enough to absolute zero only translational modes are occupied (sorry I know this is the classical forum) but at higher temperatures non-translational degrees of freedom get populated; not all of it goes into kinetic energy. Is this right?

Urshilikai said:
This is an odd question. I was talking with a social scientist the other day and he seemed to think that the amount of energy it takes to raise the temperature 1 degree Celsius or Kelvin was minutely different depending on the temperature (ie near absolute zero the energy it takes to increase T by 1 is different than the energy it takes to increase T by 1 at 300K). I tried explaining to him that kinetic energy is linearly proportional to temperature, but he insisted that he was right and that the scale was very slightly nonlinear (no proof or source I'm afraid, so it's been driving me nuts :/ ). I have done some searching to no avail, I turn to you for ideas. Are there any nonlinear effects that cause the energy separation between degrees K/C to be unequal?
The Kelvin and Celsius scales are defined so that a change of 1K is exactly the same as a change of 1 °C. The two scales differ by a constant offset.

However, the amount of energy needed to raise the temperature of some real substance by 1K or by 1 °C does vary with temperature. The heat capacity of a real substance is not constant. It is a function of temperature. An ideal gas -- that's an ideal. Some gases do come close to this ideal over some finite temperature range, but there is no such thing as an ideal gas. Nor an ideal solid.

Sorry, I'm a classical dude. I have no idea what translational modes are..

Substances undergo phase-change, in which a substance absorbs energy but the temperature doesn't rise, at the critical points of a substance (where it freezes/sublimates/melts/evaporates/etc). Once the substance has received enough energy to change phase, it starts raising in temperature again with additional energy absorbed.

D H said:
The Kelvin and Celsius scales are defined so that a change of 1K is exactly the same as a change of 1 °C. The two scales differ by a constant offset.

However, the amount of energy needed to raise the temperature of some real substance by 1K or by 1 °C does vary with temperature. The heat capacity of a real substance is not constant. It is a function of temperature. An ideal gas -- that's an ideal. Some gases do come close to this ideal over some finite temperature range, but there is no such thing as an ideal gas. Nor an ideal solid.

hrmm... does that mean that

$$\frac{1}{2}mv^2 = \frac{3}{2}kT$$

is not true for real substances..? or that I'm misinterpreting it?

Pythagorean said:
hrmm... does that mean that

$$\frac{1}{2}mv^2 = \frac{3}{2}kT$$

is not true for real substances..? or that I'm misinterpreting it?
Correct. That's for a monatomic ideal gas. It's not even true for a diatomic or multiatomic ideal gas.

There is no such thing as an ideal gas. Some gases come close to that ideal, some very, very close. But even the inert gases condense into liquids, and the atoms have a non-zero size.

D H said:
Correct. That's for a monatomic ideal gas. It's not even true for a diatomic or multiatomic ideal gas.

There is no such thing as an ideal gas. Some gases come close to that ideal, some very, very close. But even the inert gases condense into liquids, and the atoms have a non-zero size.
Even in the limit of ideal gas behavior, the heat capacity is a function of temperature.

Chestermiller said:
Even in the limit of ideal gas behavior, the heat capacity is a function of temperature.
No, Chester. Heat capacity is constant an ideal gas. For a monatomic ideal gas, cv=3/2 R. Look at post #6. It's right there.

D H said:
No, Chester. Heat capacity is constant an ideal gas. For a monatomic ideal gas, cv=3/2 R. Look at post #6. It's right there.

Smith and Van Ness, Introduction to Chemical Engineering Thermodynamics:

"Ideal-gas heat capacities (designated $C_p^{ig}$ and $C_v^{ig}$) are different for different gases; although functions of temperature, they are independent of pressure.
The temperature dependence may be shown graphically, as illustrated in Fig. 4.1, where $C_p^{ig}/R$ is plotted vs. temperature for argon, nitrogen, water, and carbon dioxide.

Don't worry about what a social scientist thinks he knows about temperature. The most dangerous thing about social scientists is what they know that just ain't so. He was just trying to get inside your head and you let him succeed.

The temperature scale is linear by definition. There are 100 divisions in the amount of heat required to bring water from zero degrees C (after it has just fused) to 100 degrees C (where it boils at one atmosphere). All of this was decided before anyone knew about translational modes or statistical mechanics.

Chestermiller said:
Smith and Van Ness, Introduction to Chemical Engineering Thermodynamics:

"Ideal-gas heat capacities (designated $C_p^{ig}$ and $C_v^{ig}$) are different for different gases; although functions of temperature, they are independent of pressure.
The temperature dependence may be shown graphically, as illustrated in Fig. 4.1, where $C_p^{ig}/R$ is plotted vs. temperature for argon, nitrogen, water, and carbon dioxide.
That first sentence makes no sense. Look at the post #6. It's right there, plain as day. Specific heat is constant for an ideal gas. As far as the second sentence is concerned, argon, nitrogen, water, and carbon dioxide are not ideal gases. There is no such thing. It's an ideal.

The thermochemical calorie was originally defined to be the amount of heat required to raise the temperature of a gram of water by one degree Celsius. However this definition is inadequate - which is something that your friend the sociologist knows:

http://en.wikipedia.org/wiki/Calorie

Urshilikai said:
It just occurred to me that near enough to absolute zero only translational modes are occupied (sorry I know this is the classical forum) but at higher temperatures non-translational degrees of freedom get populated; not all of it goes into kinetic energy. Is this right?
It depends. If the substance is monatomic, there are only translational modes (no rotational or vibrational modes). If the substance is diatomic or polyatomic, the heat capacity will begin to increase when it reaches temperatures where the rotational and vibrational modes start to become active. As temperature decreases rotational and vibrational modes cease to be active heat capacity decreases. When any substance approaches absolute 0, even the translational modes start to get frozen out.

Regarding the point that your sociologist friend was trying to make, was he referring to a particular substance or all matter? If he was simply saying that the heat capacity of water varies with temperature, he would be right. That has to do more with the intermolecular forces in water (ie. potential energy of molecules increases with temperature).

AM

I thought sociologists were mostly concerned with the properties of hot air

D H said:
That first sentence makes no sense. Look at the post #6. It's right there, plain as day. Specific heat is constant for an ideal gas. As far as the second sentence is concerned, argon, nitrogen, water, and carbon dioxide are not ideal gases. There is no such thing. It's an ideal.

This entire quote makes perfect sense to me. Smith and Van Ness has been used as a text in thermo courses for over fifty years, with several new editions, and the quote I gave is in complete agreement with the corresponding sections in numerous other highly regarded treatises, including Hougan and Watson, Perry's Chemical Engineers' Handbook, Transport Phenomena by Bird, Stewart, and Lightfoot.

I think what we may be dealing here is a difference in the definition used for an "ideal gas" by physicists vs engineers. In engineering we define an ideal gas as the limiting behavior of a real gas at very low pressures. Thus, an ideal gas is one whose
(a) PVT behavior is PV=nRT
(b) Cv and Cp are functions only of temperature.
(c) H and U are functions only of temperature.

Apparently, in physics, this is not the definition that is used. In order to clarify this issue, I'm going to start a new thread in which I solicit responses from both physicists and engineers as to their understanding of what an ideal gas is. I will be titling this thread "Definition of an Ideal Gas."

Chet

Chestermiller said:
Apparently, in physics, this is not the definition that is used.
Briefly, what physicists call an ideal gas, engineers call a perfect gas. I provided a more detailed answer in your thread on this side topic.

With that nomenclature discrepancy cleared away, I'll back to the original post.

Urshilikai said:
I tried explaining to him that kinetic energy is linearly proportional to temperature ...
That's only true for ideal (perfect) substances. It's not true for real substances. I'll give a couple of examples.

Example 1: A mix of ice and water.

Add energy and the temperature does not increase. Instead, some of the ice melts but the mix remains at 0 °C. The temperature doesn't start rising until all the ice has melted. In an ideal gas, the only form of energy is kinetic energy. That's not the case here, or in any real substance. There are also various forms of potential energy. The phase transition represents a change in potential energy and also in entropy.

Example 2: A diatomic gas.

Even kinetic energy is a bit problematic when dealing with real gases. An ideal gas has its kinetic energy equally partitioned amongst all possible outlets for that energy. There are only three degrees of freedom for a low temperature monatomic gas: Motion in three dimensional space. Helium is about as close to ideal (perfect) as a gas can get. Except at very low temperatures, specific heat is pretty much independent of temperature, and it's just what kinetic theory suggests, ##c_v = \frac 3 2 R##. Diatomic gases at low temperature behave similar to monatomic gases. Increase the temperature and specific heat eventually starts increasing toward ##\frac 5 2 R##. It's as if a couple of new degrees of freedom have been added -- and they have. The diatomic gas starts exhibiting two rotational degrees of freedom. Add even more energy and specific heat eventually starts increasing even further. Bottom line: The simple linear relation between energy and temperature seen in an ideal gas is not linear and not so simple even for a near-ideal diatomic gas.

Chestermiller: I moved your post on the engineer's distinction between an ideal gas and a perfect gas to your own thread. That discussion is off-topic here. The topic of this thread is ultimately "what is temperature?"

Never mind.

D H said:
Chestermiller: I moved your post on the engineer's distinction between an ideal gas and a perfect gas to your own thread. That discussion is off-topic here. The topic of this thread is ultimately "what is temperature?"
Actually, the title is misleading. It seems to me that this thread is really about whether heat capacity depends on temperature. The answer is that it depends...

AM

Andrew Mason said:
Actually, the title is misleading. It seems to me that this thread is really about whether heat capacity depends on temperature. The answer is that it depends...

AM

It is about whether the heat capacity of what one calls an ideal gas depends on temperature. If you are a physicist, you might have one answer to this question, while, if you are from some other discipline, you might have a different answer.

Chet

Chestermiller said:
It is about whether the heat capacity of what one calls an ideal gas depends on temperature. If you are a physicist, you might have one answer to this question, while, if you are from some other discipline, you might have a different answer.
No, it's not, Chet. There's nothing in the OP that talks about ideal gases. And please, stop with the comments about how different groups have different names for things. It's off topic in this thread.

The original post, which is admittedly a bit confused, asks three different questions, one implied:
• Is a temperature change of 1K always the same as a temperature change of 1 °C?
The answer to this question is yes. That's how the Kelvin and Celsius scales are defined.

• Is energy (always) linear with respect to change in temperature?
The answer to this question is no. That implied always is a killer. It's not just ideal (or perfect) gases versus real ones where that linear relation falls apart. The thermodynamic behaviors of a crystalline solid can be rather funky at very low temperatures thanks to quantum mechanics. And liquid helium? Superfunky.

• Is kinetic energy all that one needs to be concerned with when it comes to temperature?
The answer to this question is an emphatic no. Ideal substances don't go through phase transitions, undergo chemical reactions, dissociate, undergo radioactive decay, etc. Real substances do, and those changes involve some form of energy other than kinetic energy.

See what troublemakers social scientists are.

D H said:
No, it's not, Chet. There's nothing in the OP that talks about ideal gases. And please, stop with the comments about how different groups have different names for things. It's off topic in this thread.

The original post, which is admittedly a bit confused, asks three different questions, one implied:
• Is a temperature change of 1K always the same as a temperature change of 1 °C?
The answer to this question is yes. That's how the Kelvin and Celsius scales are defined.

• Is energy (always) linear with respect to change in temperature?
The answer to this question is no. That implied always is a killer. It's not just ideal (or perfect) gases versus real ones where that linear relation falls apart. The thermodynamic behaviors of a crystalline solid can be rather funky at very low temperatures thanks to quantum mechanics. And liquid helium? Superfunky.

• Is kinetic energy all that one needs to be concerned with when it comes to temperature?
The answer to this question is an emphatic no. Ideal substances don't go through phase transitions, undergo chemical reactions, dissociate, undergo radioactive decay, etc. Real substances do, and those changes involve some form of energy other than kinetic energy.

Sorry DH. I apologize again.

Chet

Perhaps the social scientist in the OP is thinking of the very slight the difference between Centigrade and Celsius/Kelvin due to the small difference between boiling and triple point...