Heat and temperature
Measuring temperature means converting it into something electrical, and there are four ways to do it. None of them is the best: each one wins in one range and loses in another, and choosing the wrong sensor ruins the measurement before it starts.
01Temperature is not heat
They are two different things, and they get confused all the time. Temperature measures the mean kinetic energy of the particles: it is a property of the state. Heat is energy in transit between two bodies at different temperatures: it exists while it flows and afterwards it ceases to exist.
A body does not “have” heat, just as a capacitor does not have current. It has internal energy, and heat is the way that energy enters or leaves.
If A is in thermal equilibrium with C, and B is too, then A and B are in equilibrium with each other. It looks obvious, and it is what makes measurement possible: the thermometer is that C. Without this property, two bodies that read the same on a thermometer would not necessarily be at the same temperature, and measuring would mean nothing.
| Scale | Zero at | Conversion | Where it is used |
|---|---|---|---|
| Celsius | Melting of water | — | Datasheets, everyday use |
| Kelvin | Absolute zero | \( T_K = T_C + 273.15 \) | Every physical formula: gases, Arrhenius, thermal noise |
| Fahrenheit | Historical convention | \( T_F = 1.8\,T_C + 32 \) | US datasheets |
The second row is not a whim. In \( PV = nRT \), in \( e^{-E_a/kT} \) and in the thermal noise \( \overline{v^2} = 4kTRB \), temperature enters in kelvin. Using Celsius there does not give a scale error: it gives nonsense.
02How much energy it takes
To raise the temperature of a body you have to give it energy, and how much depends on its mass and on what it is made of:
| Material | \( c \) in J/kg·K | \( \alpha \) in 10⁻⁶/K | Why it matters |
|---|---|---|---|
| Water | 4186 | — | By far the best coolant there is |
| Aluminum | 897 | 23 | The typical heat sink: light and cheap |
| Copper | 385 | 17 | Better conductor, more expensive and heavier |
| Silicon | 705 | 2.6 | Expands ten times less than aluminum |
| FR-4 epoxy glass | 1100 | 14 to 18 | The laminate of circuit boards |
The third column is the thermal expansion, and there lies one of the central problems of packaging: a silicon chip expands 2.6 parts per million per degree and the FR-4 board it is soldered to expands six times more. Every power-on and power-off cycle stretches and compresses the solder joints, and after thousands of cycles they crack. That is thermomechanical fatigue, and it is the cause of failure of a great many devices that “just stopped working on their own”.
03The four temperature sensors
Measuring temperature, in practice, means converting it into voltage or resistance. There are four usual ways, and none is the best at everything: each one wins in one range and loses in another.
| Sensor | What it delivers | Range | Its problem |
|---|---|---|---|
| Thermocouple type K | About 41 µV/°C | −200 to 1300 °C | Tiny signal, and it needs cold-junction compensation |
| Pt100 | 0.385 Ω/°C | −200 to 850 °C | It has to be excited, and the current heats it |
| NTC thermistor | Exponential drop | −40 to 150 °C | Very sensitive but not at all linear |
| Semiconductor junction | −2 mV/°C | −55 to 150 °C | Small range, but it is integrated on the same chip |
The last row is the one that explains why almost every microcontroller has a temperature sensor inside: the forward voltage of a junction drops about 2 mV per degree, and that was already there. The most linear is the Pt100, the most sensitive is the NTC, and the one with the widest reach is the thermocouple.
You move the temperature and each sensor responds with its own law: the Pt100 with Callendar–Van Dusen, the NTC with its B equation, the thermocouple and the junction with their slope. See which one is straight and which is not.
A thermocouple does not measure temperature: it measures the difference between its tip and the point where it connects to the instrument. That second point is the cold junction, and it has to be measured separately with another sensor —almost always an integrated junction— and added. A thermocouple thermometer always carries inside a second thermometer.
04In the lab
Measure the resistance of a thermistor between 0 and 80 °C with a water bath and a reference thermometer. Linearize with \( \ln R \) versus \( 1/T \), obtain the constant \( B \) and compare it with the one in the datasheet.
Determine the specific heat of an aluminum piece with an insulated-cup calorimeter, measuring the equilibrium temperature when it is dropped hot into cold water. Compare with the tabulated 897 J/kg·K and discuss the losses.
Measure the forward voltage of a diode at constant current while heating it with air. Verify the slope of −2 mV/°C and estimate what error a 10% variation in the bias current introduces.
05Common mistakes
- Saying that a body “has” heat. It has internal energy; heat is energy in transit.
- Using Celsius in physical formulas. In \( PV=nRT \), in Arrhenius and in thermal noise, it has to be kelvin.
- Confusing specific heat with heat capacity. One is per kilogram; the other is for the whole body.
- Forgetting latent heat. During a change of state energy is supplied and the temperature does not move.
- Ignoring cold-junction compensation. Without it a thermocouple measures differences, not temperatures.
- Assuming every sensor is linear. Only the Pt100 comes close; the NTC is not even remotely so.
06Self-assessment
What is the difference between heat and temperature?
Temperature is a property of the state; heat is energy in transit between bodies at different temperatures.
What is the Zeroth Law for?
It is what makes measurement possible: it guarantees that two bodies that read the same on a thermometer are in equilibrium with each other.
What is the resistance of a Pt100 at 100 °C?
138.5 Ω, according to Callendar–Van Dusen. At 0 °C it is 100 Ω, hence the name.
Why do solder joints crack with thermal cycling?
Because silicon and FR-4 expand differently —2.6 versus 16 parts per million per degree— and each cycle stretches the joint.
What does a thermocouple really measure?
The temperature difference between its tip and the cold junction. That is why the instrument carries a second sensor inside.
Why is water such a good coolant?
Because of its specific heat: 4186 J/kg·K, more than four times that of aluminum. It absorbs a lot of energy with little change in temperature.
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”). The chapters on temperature, thermal expansion and the first law, with many worked problems.
- David Halliday, Robert Resnick and Jearl Walker. Fundamentals of Physics. 11th ed., Wiley, 2018 (Spanish edition, “Fundamentos de física”). The classic treatment of heat, specific heat and changes of state.
- Jacob Fraden. Handbook of Modern Sensors. 5th ed., Springer, 2016. The four sensors of this unit with their physics, their equations and their signal-conditioning circuits.