Foundations — First law of thermodynamics — dU = dQ − dW, sign conventions

This page builds every symbol the parent note uses from absolute zero. We go in dependency order: each idea uses only the ones defined above it. If you meet a letter later and feel lost, it was defined here first.

0. The system — what are we even watching?

Before any symbol, we need a thing to watch. In thermodynamics that thing is a system: a chosen blob of matter — here, a gas trapped in a cylinder with a movable piston. Everything else in the universe is the surroundings.

Look at the figure: the dashed pale-yellow line is the boundary. Heat comes in from the flame below; the piston on top can slide, letting the gas push outward. Those are the only two ways energy moves — that is the entire law in one picture.

1. $P$ — pressure

Why the topic needs it: the gas does work by pushing the piston, and "push" is exactly force. To turn pushing into a number we use pressure. Units: pascals, $\text{Pa}=\text{N}/{\text{m}}^2$.

2. $V$ — volume

Why the topic needs it: when the piston moves, the space changes. That change in space is how the gas spends energy. So volume is the quantity whose change we will watch most closely.

3. $T$ — temperature

Why the topic needs it: for an ideal gas, "hotter" and "more internal energy" are the same statement. Temperature is the reader-friendly face of internal energy. See Internal energy and degrees of freedom for the deep version.

4. State, and the triplet $(P,V,T)$

Why the topic needs it: the star property of internal energy — that it depends only on the state, not on the journey — cannot even be stated without the word "state." This is the hinge of the whole note.

5. State function vs path function — the lake picture

This distinction confuses everyone, so it gets its own figure.

In the figure, both blue paths start and end at the same two dots (same state change) but sweep out different areas underneath. Internal energy $U$ (the level) ends up the same; heat $Q$ and work $W$ (the rain and the outflow, ∝ the swept areas) do not.

6. $U$ — internal energy

Why the topic needs it: this is the "stored" bucket. Heat in that doesn't leave as work must land here — raising $U$, i.e. raising $T$. The whole point of the First Law is to track this bucket. Built further in Internal energy and degrees of freedom.

7. $Q$ — heat

Why the topic needs it: heat is the input channel — the "food eaten." Without $Q$ the gas could never gain energy from outside without being pushed. Distinct from temperature: temperature is a state (how hot); heat is a transfer (energy on the move).

8. $W$ — work (done by the gas), and $dW=PdV$

Now we build the crucial formula. This needs the tool force × distance (from mechanics, Conservation of energy (mechanics)) — chosen because work is defined that way and nothing simpler captures "energy from pushing."

9. $d$ and $\Delta$ — "a tiny change" vs "a total change"

Why the topic needs it: work $W=\int PdV$ must be built slice-by-slice because $P$ varies, so we need the tiny $dV$. The final answers, though, are finite $\Delta U$, so we need both notations. $dU=dQ-dW$ is the slice-form; $\Delta U=Q-W$ is the summed-up form.

10. Sign convention — the $-$ in $dU=dQ-dW$

Every case is covered by one rule: watch the sign of $dV$. Expansion $\Rightarrow dV>0\Rightarrow W>0$; compression $\Rightarrow dV<0\Rightarrow W<0$; no volume change $\Rightarrow dV=0\Rightarrow W=0$.

Prerequisite map

Equipment checklist

Test yourself — cover the right side.

Connections

• Parent topic
• Internal energy and degrees of freedom
• Work done in thermodynamic processes (PV diagrams)
• Isothermal, adiabatic, isobaric, isochoric processes
• Heat capacities Cp and Cv
• Conservation of energy (mechanics)
• Second law of thermodynamics