First and Second Laws
Erasing a bit has a minimum energy cost that does not depend on the technology: it is kT·ln2 joules, and it comes from the Second Law. Thermodynamics puts a floor under the power consumption of a computer.
01The First Law: energy is conserved
The internal energy of a system changes in two ways, and only two: the heat that enters and the work that is done. Written like that it seems little, and it is the most general conservation principle there is.
| Process | What is held constant | Consequence |
|---|---|---|
| Isothermal | The temperature | \( \Delta U = 0 \), so \( Q = W \) |
| Isobaric | The pressure | \( W = P\Delta V \) |
| Isochoric | The volume | \( W = 0 \), all the heat goes into internal energy |
| Adiabatic | No heat exchange | \( Q = 0 \), the work comes out of \( U \) |
For electronic equipment in steady state the balance is trivial and therefore useful: there is no mechanical work and no accumulation, so all the electrical power that goes in comes out as heat. A 500 W power supply feeding a computer heats the room with 500 W, no matter what it is computing. That is the starting point of any room-cooling calculation.
02The Second Law: conserving is not enough
The First Law does not forbid a cold cup from heating up by itself by cooling the air: energy would be conserved all the same. What forbids it is the Second, which introduces a quantity that never decreases in an isolated system:
From this comes the limit that no machine can exceed. A heat engine that takes heat from the hot reservoir, at \( T_c \), and delivers it to the cold one, at \( T_f \), cannot have an efficiency greater than:
The same pair of temperatures, four different machines. The diagram shows the energy flows to scale. Look at the Peltier: it is the only one that does not come anywhere near its own Carnot limit, and there lies its whole story.
A heat sink is a passive exchanger: it moves heat to the surroundings, and for that the component has to be hotter than the air. It can never take it below ambient temperature, because that would mean heat flowing uphill by itself, which is precisely what the Second Law forbids. To go below ambient you need a machine that consumes work: a compressor or a Peltier.
03The thermodynamic cost of erasing a bit
Here thermodynamics meets computing in a surprising way. Erasing information is an irreversible process: two possible states become one, and that loss of uncertainty is a decrease in the entropy of the system. For the universe not to violate the Second Law, that entropy has to appear in the surroundings as heat:
That is about three zeptojoules. A current transistor switches spending on the order of a femtojoule, so today's electronics is about three hundred thousand times above the limit. There is room, but the room is not infinite, and that floor explains why reversible computing is researched: if an operation does not destroy information, in principle it has no minimum energy cost.
The same \( kT \) that appears here is the one in the diode equation, in Arrhenius and in thermal noise \( \overline{v^2} = 4kTRB \). It is no coincidence: in all four cases what is being measured is how much energy thermal agitation brings, and that number —26 meV at room temperature— is the one that sets the scale for almost all of electronics.
04In the lab
Power a resistor of known power rating inside an insulated container with water and record the temperature rise against time. Calculate the absorbed energy and compare it with the electrical energy delivered. Discuss where the difference went.
Power a Peltier cell with a heat sink on the hot face and measure the temperature of both faces and the electrical power consumed. Calculate the real COP and compare it with the Carnot COP for that same temperature difference.
Measure input and output power of a switching power supply with a variable load. Plot the efficiency against the load and calculate how much heat the enclosure has to remove in the worst case.
05Common mistakes
- Believing the Carnot efficiency can be reached. It requires reversible processes, which take infinite time.
- Using Celsius in the Carnot formula. The temperatures go in kelvin, and with Celsius the result means nothing.
- Thinking a heat sink cools. It only transports heat, and for that it needs a temperature difference in its favor.
- Confusing COP with efficiency. The COP of a refrigerator is greater than 1 without violating anything: it is not an efficiency.
- Believing entropy never decreases. It decreases locally all the time; what does not decrease is that of the universe.
- Assuming an efficient device does not heat up. All the power that goes in and does not come out as useful signal ends up as heat.
06Self-assessment
How much heat does a computer that consumes 500 W dissipate?
500 W. There is no mechanical work and no accumulation: all the electrical power ends up as heat.
Which quantity gives time its direction?
Entropy: it never decreases in an isolated system, and all spontaneous processes increase it.
What does the Carnot efficiency depend on?
Only on the two temperatures, in kelvin. Not on the fluid or the design.
Why can a COP be 4?
Because it is not an efficiency: it measures heat moved per unit of work, and moving heat costs less than creating it.
How much energy does it cost to erase a bit?
At least \( kT\ln 2 \), about 2.9 zeptojoules at 300 K. That is Landauer's limit.
Can you cool something below ambient with a heat sink?
No. It would require heat to flow uphill by itself. For that you need a machine that consumes work.
07Further reading
- Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers with Modern Physics. 10th ed., Cengage, 2019 (Spanish edition, “Física para ciencias e ingeniería”). First and second laws, heat engines, the Carnot cycle and entropy.
- Enrico Fermi. Thermodynamics. Dover, 1956 (Spanish edition, “Termodinámica”, Eudeba, reissue). One hundred and fifty pages that explain the whole subject. It remains unsurpassed as a first serious reading.
- Charles H. Bennett. The Thermodynamics of Computation — a Review. International Journal of Theoretical Physics, vol. 21, 1982. Where Landauer's limit comes from and what reversible computing means.