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UY1: Definition of first law of thermodynamics

State and apply the first law of thermodynamics in differential and finite forms with consistent sign conventions.

  • University Physics Year 1
On this page

Learning objectives

  • Apply heat, work, internal-energy, and first-law accounting consistently across processes.
Why this matters + quick links

This page gives the UY1 working model/result for Definition of first law of thermodynamics. You reuse it when you turn thermal intuition into calculable models (energy balances, process reasoning, and kinetic/statistical links), with careful sign/units checks.

The first law is energy conservation written for thermodynamic systems.

It is the backbone of every process calculation in this module: once you define the system and lock a sign convention (see Definitions), every step becomes “what heat and work crossed the boundary?“.

1) At a glance

  • Prerequisites: heat/work definitions; state vs path variables
  • Outcomes: use differential and integral forms; convert between work-on and work-by conventions
  • Key result: dE = δ Q + δ W ⇌ Δ E = Q + W
Common traps
  • Copying a formula without checking whether W means work on the system or work by the system.
  • Plugging W = PΔ V into Δ E = Q + W (this site uses W = -Wₒᵤₜ for PV work).
  • Treating δ Q or δ W as exact differentials/state functions (they are path variables).
  • Using -P dV for irreversible steps without checking whether P is internal pressure or external pressure Pₑₓₜ.

2) Setup (model, assumptions, sign conventions)

Sign convention (used in this module)
  • Q > 0: heat into the system.
  • W > 0: work done on the system.
  • Work done by the system is Wₒᵤₜ = -W.

For a closed system with only boundary heat and work transfer:

Then first law is always: Δ E = Q + W

Quick checks:

  • Units: E, Q, and W are energies (J).
  • Limiting case: for an isolated system, Q = 0 and W = 0, so Δ E = 0.

If only quasistatic PV work is present: δ W = -Pₑₓₜ dV For reversible steps, Pₑₓₜ = P.

3) Core derivation/explanation

Path variables add to a state change: dE = δ Q + δ W Integrate from state 1 to 2: Δ E = ∫₁²δ Q + ∫₁²δ W = Q + W

Why this step matters

This split explains a core thermodynamics idea: E depends only on state, while Q and W depend on process path. Integration is the step that converts path-level transfer into a state change between fixed endpoints.

With only reversible PV work: dE = δ Q-P dV And if process is reversible and entropy is introduced via δ Qᵣₑᵥ = T dS: dE = T dS-P dV

Alternative textbook form: Δ E = Q-Wₒᵤₜ This is equivalent because Wₒᵤₜ = -W.

4) Worked example

A gas receives 250 J heat while a stirrer does 40 J of work on it.

Δ E = Q + W = 250 + 40 = 290 J

If the same data are written using work output: Wₒᵤₜ = -40 J, Δ E = Q-Wₒᵤₜ = 250-(-40) = 290 J

  • Units check: every term in first law is energy (J).
  • Sanity check: both heat input and work input should increase internal energy, so positive Δ E is expected.

5) Practice set (with hints + answers)

  1. An isolated system has no heat/work exchange. What is Δ E?
  2. A process has Q = -120 J and W = +50 J. Find Δ E.
  3. If Wₒᵤₜ = 300 J and Δ E = -40 J, find Q.

Hints

  1. Set Q = W = 0.
  2. Add using your sign convention.
  3. Use Δ E = Q-Wₒᵤₜ or convert to W first.

Answers

  1. Δ E = 0.
  2. Δ E = -120 + 50 = -70 J.
  3. -40 = Q-300 ⇒ Q = 260 J.

6) Summary + next steps

  • First law is conservation of energy for thermodynamic systems.
  • With this module’s convention, always use Δ E = Q + W.
  • Equivalent formulas are fine only if the definition of W is stated clearly.

Next: More about internal energy Previous: Cyclic processes Back To Thermodynamics