# Specific heat capacity and temperature

Why does the specific heat capacity of an object changes with temperature?Please explain in detail.Thanks..

epenguin
Homework Helper
Gold Member
Not in detail - that's in the textbooks, but you also need the broad philosophy of it.

The answer is: it's because of quantum mechanics. That according to classical statistical mechanics specific heat should not change with temperature. And also in practice to a first approximation at reasonable easily accessible temperatures it doesn't. Which is quite fortunate as (with a bit of fudge) it gives the law of Dulong & Petit*1 which was a help in understanding the basics of molecular composition and structure of substances in the nineteenth century.

The reason it classically shouldn't change with temperature is that classically the energy is equally distributed between all the 'degrees of freedom' of any molecule. Monatomic gas molecules have just 3 degrees of freedom, i,e, they are free to translate in 3 dimensions/directions, averagely there is the same energy in each of these.*2 So they should all have the same molecular specific heats (calories or KW/mole/degree) and they do. A diatomic atom like hydrogen, H2 has these degrees and it also has the freedom to bodily rotate - think of H-H able to rotate in the plane of the screen looking from the front, and also perpendicular to that - imagine yourself looking down on H-H from the top of the screen.*2,3 So the specific heat of a diatomic gas should be 5/3 that of a monatomic one. And so it is for hydrogen - at high enough temperatures.

But not at lower temperatures. It declines to the same as monatomic gases. Evidently the molecules are not doing these rotations. The nonclassical quantum mechanical explanation is that the molecule cannot rotate at just any rate, the rotation is quantised, actually in equal steps of angular momentum, and it either has one or more of these quota of rotation or none at all. At low temperature when there is little energy to go around, classically they should rotate slowly, quantum mechanically not at all. These ideas when properly formulated mathematically give a quantitative description agreeing satisfactorily with experimental measurements of gas specific heats. These and other examples were influential in getting the first ideas of quantum mechanics accepted.

Another one, a first, was the specific heat of solids which is essentially zero at low temperatures where there is not enough energy to excite vibrations, accounted for quite well by Einstein in one of his 1905 breakthroughs. It has been said that one major preoccupation of his in his career is the accounting for how energy distributes itself between modes of motion (including in the electromagnetic field).

Another success of quantum mechanics is in giving explanations instead of fudges at the points I indicate with *asterisks. Fudge *1 If internal vibrations along bonds contributed the molar specific heat should not be the same for all substances as it roughly is. Fudges *2,3 It is in all this assumed that the monatomic atoms do not rotate, and analogously the diatomic one does not rotate around the axis joining the atoms, which cannot be justified classically.

There are still fudges in what I wrote, but this stuff is several chapters of your Physical Chemistry textbook! I just thought that some sort of philosophic overview as well as grinding though all the mathematical derivations of detail helps you see the way and the point of all that.

Last edited:
ysw mathsphys
thank you. but I still don't understand, because I'm still in AS level.

epenguin
Homework Helper
Gold Member
thank you. but I still don't understand, because I'm still in AS level.

Sometimes I think so am I. E.g. when I set myself to explain stuff I thought I had grasped time ago. I certainly can't explain more unless you say just what you don't understand.

ZapperZ
Staff Emeritus
thank you. but I still don't understand, because I'm still in AS level.

This is a good example to remind everyone, especially the new members, that when you ask a question that is very terse, such as this:

Why does the specific heat capacity of an object changes with temperature?Please explain in detail.Thanks..

.. all of us here have no clue on what your background knowledge is and what you are capable of understanding. If you do not put in some effort into presenting not only the question, but also what you educational level is, then some of us may end up wasting our time explaining things at a level that you do not understand, as what has happened here.

For example, what exactly "in detail" means? For many of us, it means going to as close to a First Principle explanation as one can get. In this particular case, it means going back to a microscopic origin. Yet, if you can't understand that, why are you asking for it in detail? What is "in detail" level, especially when none of us know what you are capable of understanding?

See the problem now?

Zz.

epenguin
Homework Helper
Gold Member
Quite. Though actually I did assume the questioner was about that level, as you wouldn't ask it if you were at a lower or a higher level. I assume he knows 'heat is energy'... but don't know what he doesn't understand, and exactly what he doesn't know.

It is not as annoying as putting effort into an answer and the questioner never comes back.:grumpy: I am beginning to keep a black book of those

Ken G
Gold Member
Bear in mind, a well-reasoned answer, citing basic principles, is often of value to more than just the person who asked the question, but also to anyone who is interested enough to read the answer-- regardless of how advanced is their education! I found your answer very illuminating, epenguin, expressly because it was at the AS level, even though I should be well past that by now. I don't think we can ever really learn something that is pitched at the level we are at, this is the paradox of education. That's why the OPer didn't understand, it was pitched at the level of their confusion rather than the level of what they understand. The only things you can really explain to people are things that they thought they already understood!

Oh. Sorry, I should have mentioned that I'm at AS-level.

So, can you explain to me at a level that I can understand.I know that specific heat capacity is the amount of heat needed to make a substance of 1Kg rise 1 degree celcius.
The only thing I might have understood from your answer is that at low temperature, the atoms can't rotate or something,so heat can't be passed on.
On my book, it says that when their is a small change in temperature, the specific heat capacity is almost constant. But over a wide range of temperature, it might change.Why?

Ken G
Gold Member
The only thing I might have understood from your answer is that at low temperature, the atoms can't rotate or something,so heat can't be passed on.
I think that means you pretty much understood the key thing.
On my book, it says that when their is a small change in temperature, the specific heat capacity is almost constant. But over a wide range of temperature, it might change.Why?
Imagine you are draining water out of a pool. At first, the water level drops by a fixed amount for a given amount of water you take out. That's like when the heat capacity is staying constant, the T drops a fixed amount for given heat taken out. But underneath the water of the pool, there might be structure that you don't realize when you just look at the pool. For one thing, you might have a "shallow end", such that if you drain enough water out, the shallow end starts to completely drain out. Now you will find that for that same given amount of water (energy) removed, the water level (temperature) has dropped a lot more than it did in the beginning, because now it's all coming out of the "deep end" (modes that can still actively hold energy even at low T). The presence of the "shallow end" in the pool, which is responsible for this drop in heat capacity when the T drops, is due to quantum mechanics, that's what epenguin was saying.

Another thing that can happen is, you put water into the pool, and it starts to overflow the edges. Now you can't get the water level to rise much at all, it just fills the pool and overflows if you add more water. That's like a phase transition, say when you melt ice. The more heat you put in, the more ice turns into water, but you can't get the T to rise so the heat capacity is effectively infinite in a phase transition. So the bottom line is, changes in heat capacity represent some kind of change in how the system is behaving, it's like the differences between a pool, or a pool with an empty shallow end, or a pool that is overflowing-- you usually don't notice those features unless you change the amount of water in the pool by a lot, and you usually don't notice those changes in matter when you add or subtract only a little heat, you have to add or subtract a lot.

Last edited:
epenguin
Homework Helper
Gold Member
OK one bit at a time, we may welcome other contributions as I can't be here all the time.

Yes specific heat is the amount of heat needed to make 1Kg rise 1 degree Celsius. But actually it is better to understand the explanation all round to take the term I used - molar specific heat, which is that amount needed to make 1 mole of a substance rise 1 degree.

Do you know what a mole is?

No,I don't do Chemistry.Anyway, thanks to epenguin and Kan G.