How is temperature, speed, rms speed, average speed, and kinetic energy related? Also, do collisions conserve total speed the way the conserve total kinetic energy?

Question

I've been looking at distributions of molecular speeds and all these terms have been confusing me. Speed and kinetic energy confuse me especially; how does temperature influence them? Any response would be appreciated and no pressure to answer both questions! Thank you!!!

Answer 1

I'll try to address your main question. Speed and kinetic energy. Speed is how fast an object moves. It has units are distance per unit time. Kinetic energy (KE) is the energy an object has due to its motion. Units are usually in joules, and joules are kg*m²/s². Note that these units have mass*meters in the numerator and seconds in the denominator. The KE is related to speed by the formula KE = 1/2 mv² so you can see KE is related to the square of the velocity. A small change in velocity results in a large change in KE.

Increasing the temperature will increases the speed of the particles, thus also increasing the KE.

Hopefully someone else will weigh in on the other aspects of your question.

Answer 2

1. Temperature and kinetic energy

In kinetic theory, the temperature of a gas is directly proportional to the average kinetic energy of its molecules:

⟨KE⟩ = (3/2)RT (per mole)

So when the temperature increases, the molecules have more kinetic energy on average.

2. Different ways of describing molecular speed

Because molecules move at many different speeds, we describe the distribution in a few different ways:‰

1. Most probable speed vmpvmp​: the speed at which the largest number of molecules are found (the peak of the distribution).
2. Average speed vavgvavg​: the arithmetic mean of all molecular speeds.
3. Root mean square speed vrmsvrms​: the square root of the average of the squared speeds; this value is directly connected to the average kinetic energy.

Their formulas are:

v_mp = √[(2RT)/M], v_avg = √[(8RT)/(ΠM)], v_rms = √[(3RT)/M]

All of them increase if the temperature increases, and all decrease if the molar mass increases.

3. Relationship between speed and kinetic energy

For a single molecule, kinetic energy is

KE = (1/2)mv²

This means speed is not directly proportional to kinetic energy, but to the square root of it. So doubling the speed would quadruple the kinetic energy.

4. Collisions: what is conserved?

In an ideal gas, collisions are treated as elastic:

1. Total kinetic energy is conserved.
2. Total momentum is conserved.
3. The sum of individual speeds is not conserved. One particle may slow down while another speeds up, but the total kinetic energy of the system remains constant.

In summary:

Temperature measures the average kinetic energy of gas molecules. That energy is connected to different definitions of molecular speed (most probable, average, and rms). Higher temperature means higher speeds across all these measures. In collisions, kinetic energy and momentum are conserved, but not the sum of the raw speeds.

Answer 3

Your question addresses some very interesting physics and it is no shame to find them confusing, most physicists ponder these things to the ends of their careers. An in depth answer would fill a few books, but I can touch on a couple of aspects here. It sounds like you are focusing on kinetic theory of gases, but some fundamentals may be useful.

The kinetic theory of gases relies on the notion (particularly for so-called ideal gases) that molecules interact only through binary collisions (i.e. no accounting for longer range interactions or collisions of more than 2 molecules at a time) and the collisions are elastic. Also assumed is that the molecules are very small with respect to the physical dimensions containing them, and there are very many molecules, so that statistical methods can be used with confidence. Although the assumptions sound very restrictive, they give a good approximation for many useful applications, particularly relatively dilute gases. A result of the assumptions is that the average translational kinetic energy of molecules in random motion in the gas is directly proportional to the absolute temperature of the gas. For an ideal gas that works out to 1/2 m < v² > = 3/2 kT, where the bracket means average over all the molecules in the gas and k is the Boltzmann constant. This also serves as one of the two definitions of temperature, called the kinetic definition. (The other definition is from the flow of heat between two macroscopic bodies and is called the thermodynamic definition.) Maxwell and Boltzmann worked out the statistical distribution of the velocities of ideal gas molecules in 3D, which now bears their name and from which all of the quantities you mentioned in the first part of your question can be obtained by straightforward calculations. You can look that up in many places (Wiki has a nice page on the Maxwell-Boltzmann distribution) but the key is the set of assumptions that define an ideal gas. Real gases with internal structures or very dense gases, or gases not in equilibrium with surroundings all require special treatment, but that is beyond what I think you are asking about.

Your second question seems a bit strange and it may be because you are immersed in kinetic gas theory. But the theory is based on the Newtonian notion of collisions of point particles, and in particular elastic collisions. In general Newtonian mechanics, speed and energy are not always conserved, you have to be more specific. The conserved quantities are momentum (mv in Newtonian physics) and kinetic energy, 1/2 mv² (only if the collision is elastic, in fact conservation of kinetic energy (KE) defines an elastic collision). Inelastic collisions do not conserve kinetic energy, but do conserve momentum. In an inelastic collision, some kinetic energy is converted to other forms of energy (heat, potential energy, etc) and the total energy is still conserved, just not the KE. The classic undergraduate introductory problem to inelastic collisions is two equal mass colliding rail cars, one at rest, the other moving, and when the moving car hits the stationary one, they stick together and continue on with the same momentum, but with half of the original velocity. KE is not conserved but momentum is. To get the conservation of total energy you have to figure out how much energy went into the "sticking" which may involve springs or heating, etc.

So be careful with collisions and which quantities are conserved. Even in relativity momentum and energy are conserved but with an added wrinkle that energy and mass can interchange. The momentum represents the conservation in the spatial dimensions and the energy represents the conservation in the time dimension, but that is a whole other fun discussion. Just be aware that very fast moving particles (say half the speed of light or so) will have collision results that are very different from what would be predicted with Newtonian mechanics. We could talk about that but it would be a whole other fun conversation......