Unizor - Physics4Teens - Energy - Temperature and Kinetic Energy
Channel: Zor Shekhtman on UNIZOR Education
Date: 17/04/2019

Unizor - Physics through a Looking Glass of Mathematician

Heat, Temperature & Kinetic Energy of Molecules

The goal of this lecture is to demonstrate that temperature, as a measure of heat, is proportional to average of a square of molecular speed.

Temperature is what we see on a thermometer that measures the intensity of the molecular movement. Let's examine this process of measurement using the classical mercury-based thermometer.

The thermometer shows different values of temperature based on physical property of mercury to expand, as its molecules are moving faster. The reason for this is that, as the molecules move faster, they cover longer distance before they are stopped by collision with other molecules or the walls or by a surface tension forces. The faster molecules push against obstacles with greater forces, and that is the reason why the volume of hot mercury is larger than the volume of the same mass of cold mercury.

Let's quantify this increase in volume.
There is a force of resistance that acts against the movement of the molecule, as it pushes its way through, that stops this movement. In some way it's analogous to throwing a stone vertically up against the force of gravity. The force of gravity eventually stops the movement upward, and the stone falls back on the ground.

The height of the trajectory of a stone depends on the initial speed it has at the beginning of movement on the ground level. Assume, the initial speed is V. Then the kinetic energy in the beginning of motion is M·V²/2, where M is a mass of a stone. The potential energy relative to the ground in the beginning of motion is zero.
At the highest point at height H the kinetic energy is zero but potential is M·g·H, where g is a free fall acceleration.

As we know, the total mechanical energy is constant in this case, from which follows that kinetic energy in the beginning of motion (with potential energy zero) equals to potential energy at the highest point (with kinetic energy zero):
Ekin = M·V²/2 = M·g·H
From this follows
H = V²/(2g)
So, the height of the rise and the square of initial velocity are proportional to each other, independently of mass.

Obviously, one mercury molecule does not do much, but when the heat is relatively evenly distributed among all molecules, their combined efforts push the surface of the mercury in the tube sufficiently noticeably, and the height it rises is proportional to average of a square of molecular speed.

The thermometers are different, their design and construction differ, they all must be calibrated, but one thing remains - a principle of proportionality between geometrical expansion (that is, temperature, as we read it on a thermometer's scale) and an average of squares of the molecular speed.

It is important to point out that this proportionality should be understood in absolute terms. That is, if the speed is zero, the temperature is zero as well. So, we are talking about temperature scales with zero representing absolute zero on the Kelvin's scale, like in space far from any source of energy.
The coefficient of proportionality, of course, depends on the units of measurement.

So, if the average speed of molecules doubles, the temperature on the Kelvin's scale rises by a factor of 4.

Inversely, if an object is at temperature 0°K (absolute zero), the kinetic energy of its molecule is zero, that is no molecular movements.
If we observe that the temperature has risen by a factor of 4, we can safely assume that the average speed of molecules doubles.