Temperature (T)

Absolute zero. Nothing happens at absolute zero — at least nothing that thermodynamics cares about. Yes, electrons continue their motion and chemical and nuclear energy remain non-zero. But the atoms themselves are effectively immobile, thermal energy is non-existent, and thermodynamics is meaningless.

Raise the temperature one degree above absolute zero and everything changes. The atoms begin to move. And once atoms move, they can be arranged in more than one way — different positions, different speeds, different directions. In Boltzmann’s world, there is more than one way to put this system together; entropy is greater than zero. Thermodynamics takes on meaning. Temperature is where the story begins.

What happens when temperature is raised?

At absolute zero, atoms in a solid sit at the bottom of the potential energy well where the attractive and repulsive forces are balanced and motion does not exist. As energy is transferred to the system from higher-temperature surroundings, the internal atoms start moving and temperature increases. The movement is up and down, back and forth along each side of the potential energy well. These motions are quantized; only certain motions are allowed. Each degree of temperature increase enables the atoms to access higher energy levels of motion, thus increasing entropy.

At a certain temperature, unique to the system’s composition, the back and forth motion of the atoms is strong enough for them to break free of neighboring atoms into the liquid state. As more energy flows into the system, temperature remains constant at this phase transition. The temperature can’t increase because the added energy is taken up entirely by the atoms that liquefy. More precisely, at the melting point any atom that gains enough energy to escape the solid does so immediately, preventing that energy from accumulating as a temperature rise. The thermal energy is transformed into an increase in potential energy of the atoms that liquefy. Note that the temperature of the liquid is the same as the temperature of the solid from which it came, and yet the energy of the liquid is higher on account of that higher potential energy. This phase transition results in a further increase in entropy, not from temperature since this doesn’t change but instead from the fact that atoms can now populate higher potential energy states and access a greater range of locations at the same temperature. Entropy increases here not through the kinetic energy distribution but through the location distribution. Recall that entropy is based on both atomic momentum and location.

Additional energy flowing into the system, now all liquid, increases temperature further and eventually induces another phase change, from liquid to vapor. The same isothermal condition applies to that transition as well.

In short, the addition of energy to a system starting at absolute zero increases temperature and causes phase change, increasing both energy and entropy along the way. All of this — the motion, the phase transitions, the increasing entropy — traces back to a single measurable quantity. Temperature. Let’s start there.

What is temperature?

Temperature is directly proportional to the average translational kinetic energy of the atoms and molecules in a given system. As energy is transferred into a system, the atoms move faster — assuming no phase change — their kinetic energy increases, and temperature rises. Rudolf Clausius demonstrated this connection in his derivation of the ideal gas law (here) based on first principles. This makes sense intuitively. The higher the kinetic energy, the stronger the impact of the molecules against the temperature-measuring device, and the higher the temperature reading.

When a thermometer is placed inside a system and thus becomes part of it, equilibrium is reached when the temperature of the thermometer equals the temperature of the system. Thus the reading on the meter reflects a true property of the system. While different temperature scales evolved based on this logic — Celsius, Fahrenheit — the existence of absolute zero as the lower bound established an absolute scale based on degrees Kelvin.

In a mixture of different atoms and molecules, the average translational kinetic energy per particle (1/2 mv²) is the same for all species at equilibrium. This results from the cumulative effect of many collisions rather than from any single collision. As a direct consequence, molecules with higher mass move slower than molecules of lower mass at the same temperature.

Now you might ask — what about multi-atom molecules that are twisting and turning and vibrating? How do they affect temperature? The short answer is that they don’t. They affect internal energy, which we’ll get to, but not temperature. Temperature is defined by the translational motion of the entire molecule as it moves through space. The internal motions — rotation, vibration — can be disregarded when it comes to temperature, as their net motion in any specific direction is zero [1]. The translational motion of the molecule is what matters here. As we’ll see later, when it comes to energy, however, the argument changes.

Why is a measure of translational motion so important? Because without such motion, no collisions would happen, no diffusion would happen, no heat transfer would happen, no chemistry would happen. Temperature is a direct measure of this motion. This is why we start with temperature.

What is pressure (P)?

Pressure, like temperature, is a direct consequence of atomic motion. Per Newton’s classical mechanics, pressure is force per unit area — P = F/A. Force, in turn, is the rate of change of momentum — F = d(mv)/dt = ma — the classic 2nd Law of Motion.

At the atomic level, pressure is what you get when vast numbers of atoms collide with a surface. Each collision delivers a tiny impulse — a change in momentum — to the wall. The accumulation of billions upon billions of such impulses per second per unit area is what we measure as pressure. Air molecules at room temperature are moving at roughly 500 meters per second. There are on the order of 10¹⁹ of them in a cubic centimeter. The pressure you feel against your eardrums right now is the net result of that relentless atomic bombardment by these many fast-moving atoms.

In the derivation of the ideal gas law, this physical picture translates directly into mathematics:

P = (change in momentum per collision) × (number of collisions per unit time per unit area)

Completing the derivation (here), pressure is proportional to the average kinetic energy of the atoms — and thus to temperature — multiplied by the number density of the atoms. The result is the ideal gas law: P = (n/V)RT.

This makes physical sense on both counts. Raise the temperature and the atoms move faster, hitting harder and more frequently — pressure rises. Increase the number of atoms in the same volume and there are more collisions per unit area per unit time — pressure rises again. Reduce the volume at constant temperature and the same number of atoms now collide with a smaller area more often — pressure rises once more. Every variation of the ideal gas law traces back to this single physical picture: atoms in motion, striking surfaces, transferring momentum.

Volume (V), mass (m), and weight (w)

The remaining directly measurable properties of any system are volume and mass.

Volume is the space available to the atoms. It sets the stage for everything else — governing how frequently atoms collide with each other and with the walls, how strongly pressure builds at a given temperature, and how many locations are accessible to the atoms at equilibrium. More volume means more accessible locations, more microstates, and higher entropy. Less volume means the opposite. The ideal gas law already captured this: at constant temperature, pressure and volume are inversely related precisely because squeezing atoms into a smaller space increases the frequency of wall collisions.

Mass is the quantity of matter present. As a conserved quantity, it plays a central role in the mass and energy balance (here).

At the atomic level, mass and speed are linked — heavier atoms move slower than lighter atoms at the same temperature, a direct consequence of the equal average kinetic energy established in the temperature section above. This link, not mass itself, is what governs behavior in a gravitational field. Recall that in free fall, mass is irrelevant — all bodies accelerate identically regardless of mass. What matters is vertical speed. Lighter atoms, moving faster at a given temperature, are more likely to attain the vertical speeds needed to rise higher against gravity. This is why lighter species tend to be found higher in a column of gas, and why the composition of our atmosphere changes with altitude — not because gravity acts differently on different masses, but because mass determines speed, and speed determines how high an atom can go.

Weight is mass acted upon by gravity — W = mg. While weight plays no direct role in most thermodynamic calculations, gravity acting on atomic masses produces density gradients, drives sedimentation and atmospheric stratification, and enables gravitational potential energy to do useful work, as in a waterfall.

Knowing the molecular weight of the atoms and molecules involved, mass converts naturally to number of particles. The mole — 6.022 × 10²³ entities — is simply a counting unit scaled so that atomic masses translate conveniently to laboratory quantities. Thermodynamic properties expressed per mole connect directly to the per-atom behavior that underlies them. Composition follows naturally, expressed as type and weight fractions or mole fractions as the calculation requires.

References

[1] Feynman, Richard Phillips, Robert B. Leighton, Matthew L. Sands, and Richard Phillips Feynman. 1989a. The Feynman Lectures on Physics.  Volume I.  Mainly Mechanics, Radiation, and Heat. Vol. 1. The Feynman Lectures on Physics 1. Redwood City, Calif.: Addison-Wesley. p. 39-11.

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