First Law of Thermodynamics

Equivalent forms

dU = \delta Q - \delta W

Q = \Delta U + W

\Delta U = Q - P\Delta V

The universe's bookkeeping rule — energy is never created or destroyed, only transferred or transformed.

Symbol	Name	SI unit	Dimension	Range
$Delta_U$	Change in internal energyoutput Net change in the system's internal energy	J	M·L²·T⁻²	-1000 – 1000
$Q$	Heat added Heat energy transferred into the system	J	M·L²·T⁻²	-1000 – 1000
$W$	Work done by system Work done by the system on its surroundings	J	M·L²·T⁻²	-1000 – 1000

Unit systems

Symbol	SI	CGS	Imperial
$Delta_U$	J	erg	BTU
$Q$	J	erg	BTU
$W$	J	erg	BTU

Where it holds

Universal law with no known exceptions. Applies to all thermodynamic systems. Sign conventions vary by textbook (physics:

W =

work by system; engineering:

W =

work on system).

Dimensional analysis

$[\Delta U] = [Q] - [W] \to M\cdot L^{2}\cdot T^{-2} = M\cdot L^{2}\cdot T^{-2} - M\cdot L^{2}\cdot T^{-2} \checkmark (all$ terms are energy in joules)

Discovery

Rudolf Clausius · 1850

Clausius formalized the relationship between heat and work, building on Joule's experiments showing mechanical equivalence of heat and Mayer's earlier theoretical work.

Try this

If you pump a bicycle tire, why does the pump get hot?

A gas absorbs 500 J of heat and does 200 J of work on a piston. Find the change in internal energy.

Research status: stable

Real-world applications

Engine cycle analysis (Otto, Diesel, Rankine cycles)
Refrigerator and heat pump efficiency calculations
Metabolic energy accounting in biology
Chemical reactor energy balance in industrial processes

Common misconceptions

Heat and work are not properties of a system — they are processes (path-dependent), while internal energy U is a state function
Sign convention varies: physics uses $W =$ work BY system $(\Delta U = Q - W)$ , engineering often uses $W =$ work ON system $(\Delta U = Q + W)$
Temperature change does not always mean heat was added — adiabatic compression raises temperature with $Q = 0$

Experimental verification

Joule's paddle-wheel experiment (1845) measured the mechanical equivalent of heat: 4.186 J/cal. Modern calorimetry confirms energy conservation to parts per billion in nuclear reactions

(E = mc^{2})

and chemical reactions.

Derivation

From conservation of energy: the total energy of an isolated system is constant.

For a closed system exchanging heat Q and doing work W on surroundings:

\Delta U = Q - W

.

This is an axiom (not derived from deeper principles) but validated by Joule's paddle-wheel experiment showing

1\,\mathrm{cal} = 4.186\,\mathrm{J}

.

Limiting cases

Q = 0

(adiabatic)⟶

\Delta U = -W

No heat exchange: work done by gas decreases internal energy.

W = 0

(isochoric)⟶

\Delta U = Q

Constant volume: all heat goes into internal energy.

\Delta U = 0

(isothermal, ideal gas)⟶

Q = W

Temperature unchanged: all heat input becomes work output.

What if…

What if

Q = W

(isothermal process for an ideal gas)?

Then $\Delta U = 0$ . All heat input becomes work output. The gas expands at constant temperature — its internal energy doesn't change because U depends only on T for an ideal gas.

What if the system is perfectly insulated

(Q = 0)

?

$\Delta U = -W$ . Any work done by the gas comes entirely from its internal energy, cooling it down. This is how adiabatic cooling works in expanding gases (e.g., canned air getting cold when sprayed).

1

Bicycle pump heating

Given ·

Q:: 500
W:: 200

Find · Delta_U

Steps

Identify: $Q = 500\,\mathrm{J}$ (heat absorbed), $W = 200\,\mathrm{J}$ (work done by gas)
Apply first law: $\Delta U = Q - W$
$\Delta U = 500 - 200 = 300\,\mathrm{J}$ (internal energy increases)

Result ·

\Delta U = Q - W = 500 - 200 = 300\,\mathrm{J}

2

Adiabatic compression

Given ·

Q:: 0
W:: -150

Find · Delta_U

Steps

Adiabatic means $Q = 0$ (no heat exchange)
Work is done ON the gas, so $W = -150\,\mathrm{J}$ (negative because surroundings do work on system)
$\Delta U = 0 - (-150) = 150\,\mathrm{J}$ — internal energy (and temperature) rises

Result ·

\Delta U = Q - W = 0 - (-150) = 150\,\mathrm{J}

Ideal Gas Law→Carnot Efficiency→Entropy Change→