How a Refrigerator Moves Heat Uphill

Equivalent forms

Q_h = Q_c + W

\text{COP}_{\text{heat pump}} = \frac{Q_h}{W} = \text{COP}_{\text{ref}} + 1

Run a Carnot engine backwards and 'efficiency' becomes COP — which can exceed 1 because you're moving heat, not creating it.

Symbol	Name	SI unit	Dimension	Range
$COP$	Coefficient of performanceoutput Heat removed per unit work; can exceed 1	dimensionless	1	1 – 20
$Q_c$	Heat extracted Heat pulled from the cold interior	J	M L² T⁻²	0 – 1000
$W$	Work input Electrical work driving the compressor	J	M L² T⁻²	20 – 120
$T_c$	Cold temperature Interior temperature	K	Θ	250 – 290
$T_h$	Hot temperature Room/coil temperature	K	Θ	295 – 330

Unit systems

SI:: Q, W in J; T in K
natural:: COP dimensionless
CGS:: Q, W in erg

Where it holds

COP bound is the reversible (Carnot) limit; real vapor-compression fridges reach 30–60% of it due to irreversibilities in compression and heat exchange.

Discovery

Jacob Perkins / William Cullen (principles); Sadi Carnot (limit) · 1834

Perkins patented the first vapor-compression refrigeration cycle in 1834; Carnot's 1824 reversible-engine argument set the unbeatable efficiency limit it can only approach.

Try this

How can a fridge make the inside colder than the room?

Heat naturally flows hot → cold, so a fridge seems to break the Second Law. It doesn't: it pays a 'tax' of work, and the entropy dumped to the room always exceeds the entropy removed inside.

Research status: stable

Common misconceptions

A fridge does not 'create cold'; it moves heat out, dumping more heat (Q_c + W) into the room than it removed — leaving the door open warms the kitchen.

Derivation

Energy conservation:

Q_h = Q_c + W

.

For a reversible cycle the Clausius equality gives

Q_c/T_c = Q_h/T_h

.

Eliminating Q_h:

W = Q_c(T_h/T_c - 1)

, so

COP = Q_c/W = T_c/(T_h - T_c)

.

Any real cycle has COP below this because it produces extra entropy.

Limiting cases

T_c \to T_h

⟶

COP \to \infty

: almost free to move heat across a vanishing gap

T_c \to 0

⟶

COP \to 0

: pumping heat out of a very cold space costs enormous work (Third Law)

Heat-pump mode⟶

COP_hp = COP_ref + 1

, always greater than 1 — why heat pumps beat resistive heating

Carnot Efficiency→Second Law of Thermodynamics→Entropy Change→Joule-Thomson Effect→