Foundations — Dalton's law of partial pressures

Before you can trust a single formula in the parent note, you must be able to read every letter in it. This page takes each symbol Dalton's law leans on — $P$, $V$, $T$, $n$, $R$, $p_i$, $x_i$ — and builds it from a picture. We go slowly. If you know all of these already, jump to the parent note; otherwise, read top to bottom.

1. What is a gas, as a picture?

Forget formulas for a moment. A gas is a huge number of tiny particles (molecules) flying in straight lines, occasionally bouncing off each other and off the container walls. They are mostly empty space between them.

This is exactly the model of Kinetic Theory of Gases. We only need its cartoon here.

2. Pressure $P$ — "push per unit area"

When a ball hits a wall and bounces back, it shoves the wall a tiny bit. Millions of balls hitting every second add up to a steady push. Spread that push over the area of the wall, and you get pressure.

Units you will meet:

• atmosphere (atm) — roughly the air pressure at sea level.
• millimetre of mercury (mmHg) — how tall a column of mercury the pressure can hold up; $760\text{ mmHg}=1\text{ atm}$.

3. Volume $V$ — the size of the box

The picture: the box that holds the swarm. Two ideas about volume matter for Dalton's law:

1. A gas fills its whole container — molecules spread out until they hit walls. There is no "unfilled corner."
2. In a mixture, every gas occupies the same full volume $V$. Red balls and blue balls share the whole box; neither is squeezed into a corner.

4. Temperature $T$ — how fast the balls fly

Kelvin, not Celsius — and why it must be kelvin.

Absolute zero ($0\text{ K}$) is where molecular motion stops entirely. Because $T$ appears as a multiplier in $PV=nRT$, doubling $T$ must double the push — but that only works from a scale that starts at "no motion at all." Celsius starts at the freezing point of water, which is arbitrary, so it would give nonsense (e.g. dividing by $0^{\circ }$C).

5. Amount $n$ — counting molecules by the mole

You cannot count $10^{23}$ balls one by one, so chemists bundle them.

The picture: $n$ is just "how many balls" repackaged into convenient units. More moles = more balls = more bangs = more pressure (at fixed $V,T$).

6. The gas constant $R$ — the conversion glue

You don't derive $R$; you look it up. Its job is purely to balance units. The specific value $0.0821$ is chosen so that pressure comes out in atm and volume in litres — the units used throughout the parent's examples.

7. The ideal gas equation — assembling the symbols

Now that $P$, $V$, $n$, $R$, $T$ each have a picture, the master equation reads like a sentence.

This is the Ideal Gas Equation. Every partial-pressure formula in the parent note is this equation applied to one gas at a time. That is why we built its five symbols first.

8. Partial pressure $p_i$ — the subscript that means "just this gas"

Lowercase $p$ (partial) versus uppercase $P$ (total) is a deliberate convention: small $p$ = one gas's share, big $P$ = everyone together.

9. Mole fraction $x_i$ — "your share of the crowd"

Two facts you must feel, not just memorise:

• $x_i$ is always between $0$ and $1$ (you can't have a negative share or more than everything).
• The shares add to one: ${\sum }_i{x}_i=1$ — everyone's slice together is the whole pie.

That second fact is why the partial pressures automatically sum back to the total: multiply "shares that add to 1" by $P_{\text{total}}$ and you recover $P_{\text{total}}$. This is the machinery of Mole Fraction and Concentration Terms.

10. Aqueous tension — the one extra idea for "gas over water"

When a gas bubbles up through water, some water evaporates into it. That water vapour has its own partial pressure, called aqueous tension.

This is a special case of Vapour Pressure. You need it because the barometer over water reads gas + water vapour, and Dalton's law lets you subtract the water's share: $p_{\text{dry gas}}=P_{\text{total}}-p_{\text{water}}$

Prerequisite map

Equipment checklist

Cover the right side; if you can answer each, you are ready for the parent note.

Connections

• Ideal Gas Equation — the master equation these symbols assemble into.
• Kinetic Theory of Gases — the swarm picture behind pressure and temperature.
• Mole Fraction and Concentration Terms — the home of $x_i$.
• Vapour Pressure — background for aqueous tension.
• Real Gases and van der Waals Equation — what changes when the "no interaction" cartoon fails.