# For Physicists - Speed of heat

All,

For most of my life, I’ve been fascinated by physics and how the world around us is ‘configured’. Though I’m not very good at math (I’m currently looking for a good place online to learn…suggestions appreciated!!), I’ve been able to hold my own on thought problems like those for special relativity (The light clock really made time dilation understandable!!).

What I’ve come to understand is that ‘heat’ as we know it, on a atomic level is the excitation/movement of atoms. A faster movement corresponds to a higher heat.

What I havent been able to find is where this is quantified; i.e. what difference in this movement corresponds to a one degree increase in temperature.

So…at 50F the speed of the atoms is x, while at 51F the speed has increased to x+?. Also, does this speed change based upon the atomic composition of the material being heated?

Also, I’d like to post questions to this thread for our resident physicists to show off their skills at making these topics understandable to a troglodyte such as myself.

Thanks, in advance, for your responses and the education these responses will provide me.

``Steve``

Heat is basically a type of kinetic energy. In bulk materials, say a block of ice, we take the statistical average of that energy and call it temperature. So in that block of ice, there are molecules of water which have a lot of kinetic energy and some with very little but on average they posses X amount of kinetic energy.

Now, if you pour 1000 J of energy into a gas and compare that with pouring 1000 J into a solid you’ll see differing end temperatures. A gas has few constraints on the movement of its molecules so its temperature will jump. A metal has far tighter bounds on its components so its temperature may barely move.

If you just want a formula you can use heat capacity C[sub]v[/sub]

C[sub]v[/sub]=Change in Heat / Change in Temperature.

so
T[sub]2[/sub] = T[sub]1[/sub] + Change in Heat / C[sub]v[/sub]

What you’re looking for is how a change in temperature affects the speed of the particles in a substance. It’s important to realize that temperature is a bulk property of a substance in thermodynamic equilibrium.

When a substance’s temperature is governed only by the speed of its constituent particles (which is a fair approximation for a substance at equilibrium in all but the most extreme situations), then its absolute temeperature is given by:

T = 2 ̅E[sub]KE[/sub]/3k

Where ̅E[sub]KE[/sub] is the average kinetic energy of a constituent particle and k is Boltzmann’s constant. If the particles are of the same mass then:

̅E[sub]KE[/sub] = mv[sub]rms[/sub]/2

Where m is mass and v[sub]rms[/sub] is the “root-mean-square velocity” (which is a measure of the average speed of the constituent particles) and hence:

T = (mv[sub]rms[/sub][sup]2[/sup])/(3k)

So in this simple situation, which is also often a good approximation for real substances, the temperature depends on the mass and average speed of its constituent particles.

“speed of heat” is likely related to heat capacity. I recall a computer modelling course I took a while ago. Imagine the solid item you are modelling as a collection of say, 1mm cubes stuck together. the adjacent cube(s) will only heat up when the cube they are next to has heated up. The greater the heat capacity of the substance (the more calories it takes to raise it 1C) the longer before the adjacent blocks heat up, since heat only flows when one pot is hotter than its adjacencies.

Of course, we are talking conduction with no convection or radiation involved. Different materials have different “thermal conductivity” and so will heat up at different speeds. There’s a Wikipedia page for “thermal conduction”.

Temperature is a measure of the averaged randomized movement (or potential for movement) of particles in a given medium. Heat is a measurement of the state of thermal energy (that is, the kinetic energy of the random motions of particles) in that medium. In other words, “heat” isn’t a quantity of things, it is a measure of state. The flow of heat from one system to another is governed by heat transfer in the form of radiation, convection, and conduction. The rate at which heat may be transferred depends on the mechanism for the transfer and the difference in temperature between the systems.

A good online resource to introduce this topic (and other science and math topics) is Khan Academy

Stranger

To make it a bit more obvious what the last two folks said. …

The term “speed of heat” generally refers to the speed at which heat propagates through a substance. i.e. apply energy to one end of a long rod and measure how long it takes for the other end to begin gaining temperature. That measures how fast the heat flowed through the rod.

That is very different from “what is the average vibrational speed of a molecule of iron in a block uniformly at 300K”. Which is what Asympotically fat answered in post #3.

But the speed of the individual molecules is, in fact, what the OP was asking about.

Temperature is a measure of kinetic energy per molecule. Kinetic energy is (mv**2) / 2, where m is mass and v is speed. So, for a given temperature, lighter molecules move faster than heavier ones. Also, temperature increases as the square root of the molecular speed (so an increase from 0 degrees to 1 degree involves a greater increase in speed than an increase from 100 degrees to 101 degrees).

There’s also the energy of rotation. Non-spherical molecules (like water, for instance) will pick up spin when they bounce off each other. This rotational energy also counts as heat, and contributes to temperature.

Another thing is that temperature is the average kinetic energy per molecule. Even in a substance that’s at a uniform temperature, the molecules won’t all have the same energy.

It should be possible to figure out the average speed of a molecule of a pure, monatomic element at a given temperature. Let’s take iron at 100 degrees Celsius, which is 373.15 Kelvin. The specific heat of iron is 0.450 Joules per gram per degree. One mole of iron (6.02 * 1023 molecules) is 55.845 grams. So a mole of iron at this temperature would have 373.15 * 55.845 * 0.450 ~= 9377 Joules of heat energy. This comes to about 1.558 * 10-20 Joules per molecule. A Joule is kilograms times speed squared, and the mass of an iron atom is about 9.273 * 10 ** -26 kilograms, so the average speed of an iron atom at this temperature is about 410 meters per second.