Foundations — First law - ΔU = q + w (chemist sign convention)

By the end of this page you will be able to point at every letter in the topic's central equation and say what it is, what it looks like, and why the topic can't live without it. We build the pieces in strict order — each one leans only on the ones before it, and we will not write the equation itself until every symbol in it has been earned.

1. The "system" and the "surroundings" — where the boundary is

Everything starts with drawing a line. Pick the chunk of the universe you care about — a gas in a cylinder, a reaction in a beaker. That chunk is the system. Everything else — the air, the bench, the water bath, you — is the surroundings. The imaginary skin between them is the boundary.

Why the topic needs it. Every symbol later — heat, work, stored energy — is measured relative to the system. "Positive" will mean "into the box." Without a fixed box, "in" and "out" are meaningless. This is exactly why chemists pick the chemist convention: they stand inside the box and ask "what happened to me?"

2. Internal energy $U$ — the "stuff stored inside"

Zoom into the box. The molecules are never still: they fly around (kinetic energy) and they pull/push on each other (potential energy). Add up every scrap of that microscopic energy and you get $U$, the internal energy.

That "rose or fell" is the key move. We almost never use $U$ itself; we use its change.

The Greek $\Delta$ — "change in"

The triangle $\Delta$ (Greek capital delta) is shorthand for final minus initial:

Why the topic needs $\Delta U$ and not $U$. We can measure transfers (heat in, work done) precisely, but the absolute internal energy of a mole of gas is a monstrous, unknowable number. The topic is written in $\Delta U$ because differences are all we can ever pin down — and all we ever need.

3. Heat $q$ — energy through the "random" door

Put the box next to something hotter. Fast molecules outside slam into the boundary and speed up the molecules inside — no coordinated push, just disorganized jostling driven by a temperature difference. That flow of energy is heat, written $q$.

Contact collisions are only one way this happens. Heat can also cross a boundary by radiation — infrared "light" carrying energy across empty space, the way a fire warms your face without touching it, or the way the Sun heats the Earth through vacuum. The common thread of all heat is not the mechanism but the driver: a temperature difference, and energy delivered in a disorganized, non-directed way.

Why the topic needs it. Heat is door number one. In Calorimetry and Bomb calorimetry this is literally the thing we measure. See Heat q for the deeper story.

4. Work $w$ — energy through the "organized" door

Now push the boundary with a coordinated force — a piston sliding, a plunger compressing gas. That is work: energy transferred by a directed force moving through a distance. It is the opposite of heat's randomness: every molecule is nudged the same way.

The most common flavour in chemistry is pressure–volume work, or PV work: a gas expanding against outside pressure, or being squashed by it. In general the outside pressure can change while the volume changes, so the honest statement adds up (integrates) each tiny slice of push over each tiny slice of volume swept:

Let's earn every sign in that formula — this is where students slip.

• Expansion: gas pushes the piston out, so $\Delta V>0$. The minus sign makes $w<0$: the system spent energy pushing on the world. Energy left the box.
• Compression: the piston is forced in, so $\Delta V<0$. Minus a negative is positive: $w>0$. The world poured energy into the box.
• Rigid container (bomb calorimeter): the walls can't move, so $\Delta V=0$, hence $w=0$. No PV door.

See Work w for the full treatment, including the non-PV kinds (electrical, shaft/stirring) that Mistake 4 in the parent warns about.

5. State functions vs path functions — why $U$ is special but $q,w$ are not

Here is the subtle idea that makes the whole subject work. Imagine hiking from a valley to a peak.

• Your altitude gain depends only on where you started and finished — not the trail. That's a state function. $U$ (and later H) behave exactly like altitude.
• The distance you walked depends on whether you zig-zagged or went straight. That's a path function. $q$ and $w$ behave like distance walked.

Why the topic needs this. Two different paths give two different pairs $(q,w)$ — yet they always add to the same $\Delta U$. That is the deep gift of the first law, and the reason Hess's Law lets us add reaction steps like Lego bricks.

The tiny letters $d$ and $\delta$

You'll see $dU$ but $\delta q$ and $\delta w$. This isn't decoration:

Why can two path-dependent dribbles add up to a path-independent change? Because the boundary only has those two doors. Whatever the route, every joule that changed $U$ had to enter as either heat or work — there is no third accountant. So while nature is free to split the total between $q$ and $w$ differently on each path (send more through the heat door here, more through the work door there), the two dribbles are forced to add to the same fixed total, since that total is just the difference in a stored quantity that doesn't care how you got there. The sum is pinned even though each piece floats.

6. Putting the doors together — the first law itself

Two doors, one tank. Whatever enters through the heat door plus whatever enters through the work door raises the water level. Now — and only now, with every symbol defined — we may write the equation the whole topic rests on:

Every symbol is now earned: $\Delta U$ = change in stored energy (§2), $q$ = heat through the random door (§3), $w$ = work through the organized door (§4), and the equals itself is conservation of energy applied to the box in §1. Nothing left unexplained.

Equipment checklist

Self-test: cover the right side, answer aloud, then reveal.