Refrigerators and heat pumps — COP — Physics notes

What is happening physically

Figure — Refrigerators and heat pumps — COP

Deriving the energy balance (HOW)

Defining COP from first principles

Formula COP definitions

Refrigerator (benefit = cooling $Q_C$ , cost = $W$ ): $\text{COP}_{ref}=\frac{Q_C}{W}=\frac{Q_C}{Q_H-Q_C}$

Heat pump (benefit = heating $Q_H$ , cost = $W$ ): $\text{COP}_{hp}=\frac{Q_H}{W}=\frac{Q_H}{Q_H-Q_C}$

Relation between them (derive it): divide top and bottom is unnecessary — just subtract: $\text{COP}_{hp}-\text{COP}_{ref}=\frac{Q_H-Q_C}{W}=\frac{W}{W}=1$ $\boxed{\text{COP}_{hp}=\text{COP}_{ref}+1}$ Why? Both share the same denominator $W$ . Their numerators differ by $Q_H-Q_C=W$ , so the COPs differ by exactly 1. A heat pump is always at least as good as 1 — even a terrible one gives you back the work as heat.

The best possible: Carnot (reversible) COP

Formula Reversible heat ratio → Carnot COP

For a reversible cycle, $\Delta S_{universe}=0$ : $\frac{Q_C}{T_C}=\frac{Q_H}{T_H}\quad\Rightarrow\quad \frac{Q_H}{Q_C}=\frac{T_H}{T_C}$ Why this step? Reversible heat exchange at temperature $T$ transfers entropy $Q/T$ . No net entropy change of the universe means the entropy pulled from cold equals that dumped hot.

Substitute into the COP formulas (divide num & denom by $Q_C$ or $Q_H$ ): $\text{COP}^{Carnot}_{ref}=\frac{T_C}{T_H-T_C}\qquad \text{COP}^{Carnot}_{hp}=\frac{T_H}{T_H-T_C}$ These are the maximum possible COPs for given $T_H,T_C$ ( $T$ in kelvin). Real devices have $\text{COP} < \text{COP}^{Carnot}$ .

Common mistake Steel-manned common errors

1. "Using Celsius in the Carnot formula." Feels right because we measure room temps in °C. But $Q\propto T$ comes from entropy $Q/T$ , valid only on an absolute scale. Fix: always convert to kelvin: $T(\text{K}) = T(°\text{C}) + 273$ .

2. "COP must be less than 1, like efficiency." Feels right because engine efficiency $\eta<1$ . But COP is heat moved ÷ work paid, and you can move far more heat than the work you spend — so COP is usually > 1. Fix: COP is not an efficiency; it's a leverage ratio.

3. "Refrigerator and heat pump have the same COP." They use the same machine, so it feels identical. But the numerator differs ( $Q_C$ vs $Q_H$ ). Fix: remember $\text{COP}_{hp}=\text{COP}_{ref}+1$ .

4. "Sign confusion: $W = Q_C - Q_H$ ." Fix: the device dumps more heat hot than it took cold; the surplus is the paid work, so $W=Q_H-Q_C>0$ .

Recall Feynman: explain to a 12-year-old

Heat naturally slides from hot to cold, like a ball rolling downhill. A fridge is a tiny pump that pushes the "heat-ball" back uphill — from the cold inside to the warm room. Pushing uphill takes effort, so you plug it in and pay with electricity. The COP just says: "for every push of effort I give, how many heat-balls did I carry up?" A good fridge carries many balls per push, so its COP is a big number. And a heat pump is the same machine, but now you stand in the room enjoying all the heat it dumps — including the effort you spent, which also turns into warmth!

First law for a refrigerator cycle (positive magnitudes)

Q_H = Q_C + W

, since

\Delta U_{cycle}=0

COP of a refrigerator

\text{COP}_{ref}=\dfrac{Q_C}{W}=\dfrac{Q_C}{Q_H-Q_C}

(benefit = cooling).

COP of a heat pump

\text{COP}_{hp}=\dfrac{Q_H}{W}=\dfrac{Q_H}{Q_H-Q_C}

(benefit = heating).

Relation between the two COPs

\text{COP}_{hp}=\text{COP}_{ref}+1

Carnot COP of a refrigerator

\dfrac{T_C}{T_H-T_C}

, with

T

in kelvin.

Carnot COP of a heat pump

\dfrac{T_H}{T_H-T_C}

, with

T

in kelvin.

Why is COP often greater than 1?

It is heat moved per work paid, not a fraction of energy converted; you can move more heat than the work you spend.

Reversible condition giving the heat ratio

\Delta S_{universe}=0 \Rightarrow Q_C/T_C = Q_H/T_H \Rightarrow Q_H/Q_C = T_H/T_C

Effect of a larger temperature gap on COP

COP decreases (denominator

T_H-T_C

grows).

Why a heat pump beats a 1:1 electric heater

It delivers

W

as heat AND drags extra heat

Q_C

from outside, so

\text{COP}_{hp}>1

Connections

Heat Engines and Efficiency — a refrigerator is a heat engine run in reverse; $\eta$ and COP share the same $Q_H,Q_C,W$ .
Carnot Cycle — gives the maximum COP and the entropy condition.
Second Law of Thermodynamics — forbids COP = ∞ and heat flowing cold→hot without work.
Entropy — $Q/T$ bookkeeping underlies the reversible heat ratio.
First Law of Thermodynamics — source of $Q_H=Q_C+W$ .

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, garmi hamesha hot se cold ki taraf khud chali jaati hai — jaise ball pahaad se neeche ludhakti hai. Lekin fridge ko chahiye iska ulta: andar ki thandi cheez se garmi nikaal kar bahar warm room mein daalni hai, yaani garmi ko uphill push karna hai. Yeh nature free mein nahi karta (second law), isliye humein work (bijli, compressor) kharch karna padta hai. Energy balance simple hai: $Q_H = Q_C + W$ — jitni garmi cold se nikali plus jitna work diya, utni garmi hot side dump hoti hai.

Ab COP ka matlab hai "fayda divided by kharcha". Fridge mein fayda hai cooling, toh $\text{COP}_{ref}=Q_C/W$ . Heat pump mein fayda hai heating (room garam karna), toh $\text{COP}_{hp}=Q_H/W$ . Dono same machine hain, bas point of view badalta hai — aur ek mast relation yaad rakho: $\text{COP}_{hp}=\text{COP}_{ref}+1$ .

Best possible value Carnot deta hai: $\text{COP}_{ref}=T_C/(T_H-T_C)$ aur $\text{COP}_{hp}=T_H/(T_H-T_C)$ , lekin temperature kelvin mein hi daalna — Celsius daaloge toh galat aayega. Note karo: jitna chhota temperature gap, utni badi COP, kyunki garmi ko chhoti pahaadi par chadhana sasta hai.

Important baat: COP usually 1 se zyada hoti hai, efficiency jaisi nahi. Aur isiliye heat pump winter mein simple electric heater se behtar hai — 1 J bijli se woh kai joule garmi ghar mein deta hai, kyunki bahar ki free garmi bhi andar kheench laata hai. Exam mein bas first law $Q_H=Q_C+W$ aur ye do COP formulas pakad lo, sab questions ban jayenge.

Refrigerators and heat pumps — COP

What is happening physically

Deriving the energy balance (HOW)

Defining COP from first principles

The best possible: Carnot (reversible) COP

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Thermodynamics

Connections