# Interpretation of temperature in liquids/solids

Homework Helper
TL;DR Summary
What is the physical interpretation of temperature of a mass of liquid or a solid?
Usually, the mental image of temperature is: an internal property of a bulk of matter, which typically describes the average kinetic plus rotation/vibration energy of molecules, so we imagine a gas in which temperature is a measure of how quick molecules are, and how frequently they collide one to another. The higher the temperature, the more energy of molecules.

Let's switch to a liquid at room temperature (water) or a crystalline solid (a bar of pure iron). How would you define their temperature? Would it for example be a measure of the average energy of the "electron gas" in the conduction band? What about liquid water, which lacks a rigid crystalline structure, being just a scattter of molecules kept together with vdW forces and "hydrogen bonds"?

The simplest interpretation is that "temperature is the average KE of the molecules". That may be too simple for some, but once you start considering changes of phase, it becomes complicated. It then requires a deeper energy analysis than simple temperature.

Gold Member
TL;DR Summary: What is the physical interpretation of temperature of a mass of liquid or a solid?

Usually, the mental image of temperature is: an internal property of a bulk of matter, which typically describes the average kinetic plus rotation/vibration energy of molecules, so we imagine a gas in which temperature is a measure of how quick molecules are, and how frequently they collide one to another. The higher the temperature, the more energy of molecules.

Let's switch to a liquid at room temperature (water) or a crystalline solid (a bar of pure iron). How would you define their temperature? Would it for example be a measure of the average energy of the "electron gas" in the conduction band? What about liquid water, which lacks a rigid crystalline structure, being just a scattter of molecules kept together with vdW forces and "hydrogen bonds"?
Regarding the "physical interpretation of temperature", I would rely on “An Introduction to Thermal Physics” by Daniel V. Schroeder (Oxford University Press 2021). Schroeder's proposal for a theoretical definition of temperature is:

Temperature is a measure of the tendency of an Object to spontaneously give up energy to its surroundings. When two objects are in thermal contact, the one that tends to spontaneously lose energy is at the higher temperature.

On base of this theoretical definition, one arrives at the thermodynamic definition of temperature:

"The temperature of a system is the reciprocal of the slope of its entropy vs. energy graph. The partial derivative is to be taken with the system's volume and number of particles held fixed;* more explicitly,
$$\frac 1 T\equiv\left( \frac {\partial S} {\partial U}\right)_{N,V} .$$"

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TeethWhitener, jbriggs444, berkeman and 2 others
Staff Emeritus
I think it is better to think of temperature as how it behaves, rather than a "interpretation" in terms of some other quantity. Yes, for a gas, it's average kinetic energy, but lots or things aren't gasses.

Temperature determines the direction of heat flow, from higher temperature to lower temperature,

DrClaude
Hillbillychemist
Temperature is the average kinetic energy whether it is expressed as translation as in liquids and gases or vibration in solids.

Baluncore
Mentor
Please, lets get away from the "temperature is kinetic energy" high-school narrative.

The simple picture is
Temperature is a measure of the tendency of an Object to spontaneously give up energy to its surroundings. When two objects are in thermal contact, the one that tends to spontaneously lose energy is at the higher temperature.

PhDeezNutz, Lord Jestocost, weirdoguy and 1 other person
Homework Helper
I am sorry, nondisrespect meant, but what makes you think I have not asked about the complicated phenomenological explanation?

Please, lets get away from the "temperature is kinetic energy" high-school narrative.
How is the energy transferred if it is not by kinetic interaction?

Mentor
How is the energy transferred if it is not by kinetic interaction?

Klystron and Lord Jestocost
Mentor
I am sorry, nondisrespect meant, but what makes you think I have not asked about the complicated phenomenological explanation?
My post wasn't addressing the OP. Let me do that now.
Usually, the mental image of temperature is: an internal property of a bulk of matter, which typically describes the average kinetic plus rotation/vibration energy of molecules, so we imagine a gas in which temperature is a measure of how quick molecules are, and how frequently they collide one to another. The higher the temperature, the more energy of molecules.
I would consider the energy to be the "internal property of a bulk of matter." Generally speaking, an increase in temperature mean an increase in energy, but the two things are not the same. This is exemplified by heat capacity: the amount of energy necessary for a certain change in temperature is different fro different substances. Then you also have to account for phase transitions, where a lot of energy can flow without a change in temperature.

This is why I like Schroeder's simple picture of temperature describing the tendency of a thermodynamic system to exchange energy with another.

Let's switch to a liquid at room temperature (water) or a crystalline solid (a bar of pure iron). How would you define their temperature? Would it for example be a measure of the average energy of the "electron gas" in the conduction band? What about liquid water, which lacks a rigid crystalline structure, being just a scattter of molecules kept together with vdW forces and "hydrogen bonds"?
In more complex systems than ideal gases, the energy is distributed among the different degrees of freedom.

Astronuc and Lord Jestocost
Gold Member
I think it is better to think of temperature as how it behaves, rather than a "interpretation" in terms of some other quantity. Yes, for a gas, it's average kinetic energy, but lots or things aren't gasses.

Temperature determines the direction of heat flow, from higher temperature to lower temperature,
That's not as easy as this. Heat can flow in any direction in a solid, and in particular against the direction of hot to cold.

Take Fourier's law q equals -kappa grad T. Then, remember thermoelectricity exists, so consider the generalized Fourier's law q equals -kappa grad T plus ST J. The first term is the usual Fourier's term, the latter is a Peltier heat which exists whenever there is an electric current. S can have either sign, J can have any direction, and S can even be a non symmetric tensor.

That, or consider anisotropic matetials, where kappa is a tensor. Write down Fourier's law under matricial form and you'll get that it's possible for a material to develop a transverse thermal gradient (and so a heat flux in that direction) even though it isn't the hot to cold direction.

dextercioby
Staff Emeritus
It is true that things are complicated. Negative absolute temperatures are another complication. But I think it is better to go from B-level to I-level to A-level definitions than to get to the end before the poster has reached the middle.

DrClaude and Bystander
Gold Member
Given that @dextercioby is already an SA/HH and is asking for an intermediate answer, I think really internalizing the definition from @Lord Jestocost ’s post will help immensely. Essentially, it says that if you increase the energy of a system and its entropy only goes up a little bit, then the system has a high temperature. Conversely, if you increase the energy and its entropy goes up a lot, it’s at a low temperature.

If you think of entropy as a function which counts the number of microstates of a macroscopic system, then the above definition has an intuitive interpretation: at very low temperatures, the number of possible microstates is quite low and increasing the energy of the system causes this number to increase rapidly. However, at very high temperatures, the number of possible microstates becomes very large and increasing the energy doesn’t open up as many new microstates proportionally.

Fra and DrClaude
Gold Member
Given that @dextercioby is already an SA/HH and is asking for an intermediate answer, I think really internalizing the definition from @Lord Jestocost ’s post will help immensely. Essentially, it says that if you increase the energy of a system and its entropy only goes up a little bit, then the system has a high temperature. Conversely, if you increase the energy and its entropy goes up a lot, it’s at a low temperature.

If you think of entropy as a function which counts the number of microstates of a macroscopic system, then the above definition has an intuitive interpretation: at very low temperatures, the number of possible microstates is quite low and increasing the energy of the system causes this number to increase rapidly. However, at very high temperatures, the number of possible microstates becomes very large and increasing the energy doesn’t open up as many new microstates proportionally.
Taking this a step further, we can see that heat flow from high temperature to low temperature is simply a reflection of the second law of thermodynamics. When an object at high temperature comes into contact with an object at low temperature, the energy flows in the direction in accordance with the second law ##\Delta S\geq0##. Since a change in energy is associated with a much larger change in entropy for low temperature objects than for high temperature objects, the increase in energy will have to happen to the low temperature object in order to be consistent with ##\Delta S\geq0##, so heat will flow from hot to cold.

Philip Koeck
Taking this a step further, we can see that heat flow from high temperature to low temperature is simply a reflection of the second law of thermodynamics. When an object at high temperature comes into contact with an object at low temperature, the energy flows in the direction in accordance with the second law ##\Delta S\geq0##. Since a change in energy is associated with a much larger change in entropy for low temperature objects than for high temperature objects, the increase in energy will have to happen to the low temperature object in order to be consistent with ##\Delta S\geq0##, so heat will flow from hot to cold.
Could heat flow from a colder to a warmer object if the warmer object increases the total entropy for example by evaporating.

Mentor
Could heat flow from a colder to a warmer object if the warmer object increases the total entropy for example by evaporating.
No. Of course, the warmer object will see its temperature decrease due to evaporation, and at one point might cool down below the colder object's temperature.

Lord Jestocost