Thermal Equilibrium and Temperature Scales

A2 · 11 min

Temperature is one of the most familiar quantities in physics and one of the hardest to pin down. It decides which way thermal energy flows, it can be measured with anything whose properties change when it gets hotter, and it has a true zero, absolute zero, that does not depend on any substance at all. This note covers thermal equilibrium, how thermometers work, why different thermometers can disagree, and the thermodynamic (kelvin) scale used in every A Level gas and thermodynamics calculation.

Temperature and the direction of energy flow

Put a hot cup of tea in a cold room. Energy flows from the tea to the room until they reach the same temperature. Put an ice cube in a warm drink, and energy flows from the drink into the ice. The rule is always the same.

Key result

Thermal energy is transferred from a region of higher temperature to a region of lower temperature.
Regions at the same temperature are in thermal equilibrium: there is no net transfer of thermal energy between them.

"Net" matters. In thermal equilibrium, energy is still exchanged between the two objects at the molecular level (molecules collide, radiation is emitted and absorbed), but the flows in each direction are equal.

This gives temperature its physical meaning: temperature is the property that decides whether two objects in thermal contact are in thermal equilibrium. Two objects are at the same temperature exactly when no net energy flows between them. Temperature is not the same as thermal energy: a bath of warm water contains far more internal energy than a red-hot spark, but energy flows from the spark to the bath because the spark is at the higher temperature.

Why thermometers work

If object A is in thermal equilibrium with a thermometer, and object B is also in thermal equilibrium with the same thermometer, then A and B are in thermal equilibrium with each other (this is sometimes called the zeroth law of thermodynamics). That is what lets us use a thermometer to compare temperatures at all: the thermometer reaches the temperature of whatever it is in contact with, and its reading then tells us that temperature.

A thermometer must be allowed to reach thermal equilibrium with the object before it is read. That takes time, which is why thermometers with small heat capacity respond fastest.

Thermometric properties

To measure temperature we need a physical property that varies with temperature in a known, repeatable way. Any such property can be used to make a thermometer.

Thermometric property	Thermometer	Notes
Density (and so volume) of a liquid	Liquid-in-glass (mercury or alcohol)	Cheap and direct-reading; fragile; limited range; slow response
Volume of a gas at constant pressure	Constant-pressure gas thermometer	Very wide range and accurate; bulky and slow; used as a standard
Pressure of a gas at constant volume	Constant-volume gas thermometer	Standard instrument for defining the thermodynamic scale
Resistance of a metal	Platinum resistance thermometer	Accurate, wide range, fairly linear; needs a circuit; slow (large probe)
E.m.f. of a thermocouple	Thermocouple	Very small junction so fast response and measures at a point; very wide range; can be read remotely; e.m.f. small and non-linear
Resistance of a semiconductor	Thermistor	Very sensitive over a small range; very non-linear

The syllabus specifically names four examples: the density of a liquid, the volume of a gas at constant pressure, the resistance of a metal, and the e.m.f. of a thermocouple.

A thermocouple consists of two wires of different metals (for example copper and constantan) joined at two junctions. When the junctions are at different temperatures, an e.m.f. is produced, which a high-resistance voltmeter can measure. One junction is kept at a known reference temperature (such as melting ice) and the other is placed where the temperature is to be measured.

Empirical scales and why thermometers disagree

To turn a thermometric property into a temperature scale, choose two fixed points, for example the melting point of pure ice ( $0\ ^\circ\text{C}$ ) and the boiling point of pure water at standard pressure ( $100\ ^\circ\text{C}$ ). Measure the property $X$ at each, then assume $X$ varies linearly with temperature in between:

\theta = \frac{X_\theta - X_0}{X_{100} - X_0} \times 100\ ^\circ\text{C}

A scale made like this is an empirical scale. Different thermometers agree at the fixed points by definition, but between and beyond them they generally disagree, because different properties do not vary in exactly the same (linear) way with temperature. The resistance of platinum, the e.m.f. of a thermocouple and the volume of mercury all vary slightly non-linearly, each in its own way.

The thermodynamic temperature scale

Physics needs a temperature scale that does not depend on the behaviour of any particular substance. That is the thermodynamic (or absolute, or kelvin) scale.

Key result

The thermodynamic scale of temperature does not depend on the property of any particular substance.
Its zero is absolute zero, $0\ \text{K}$ : the lowest possible temperature.
Its unit is the kelvin (K), an SI base unit. A temperature change of $1\ \text{K}$ is the same size as a change of $1\ ^\circ\text{C}$ .
Conversion:

T/\text{K} = \theta/^\circ\text{C} + 273.15

The scale is defined theoretically from thermodynamics, using absolute zero and the triple point of water (the unique temperature and pressure at which ice, liquid water and water vapour coexist), which is assigned the value $273.16\ \text{K}$ exactly. In practice, a gas thermometer using a gas at low pressure (which behaves very nearly as an ideal gas) agrees with the thermodynamic scale, so ideal-gas behaviour lets us realise the scale experimentally.

Absolute zero

At absolute zero, the internal energy of a substance is at its minimum possible value. For an ideal gas the pressure (at constant volume) or volume (at constant pressure) would become zero. Absolute zero can be approached but never actually reached.

Absolute zero is $0\ \text{K} = -273.15\ ^\circ\text{C}$ . In most calculations $273$ is accurate enough, but use $273.15$ when the data justify it.

Watch out

Temperature differences need no conversion. A rise from $20\ ^\circ\text{C}$ to $50\ ^\circ\text{C}$ is a rise of $30\ \text{K}$ . Adding $273$ to a temperature change is a classic error. Only absolute temperatures (in $pV = nRT$ , $\tfrac{3}{2}kT$ , Wien's law and so on) must be in kelvin.

Watch out

Writing "degrees kelvin". The unit is the kelvin, symbol K, with no degree sign: $300\ \text{K}$ , not $300\ ^\circ\text{K}$ .

Worked examples

Converting temperatures

Convert (a) body temperature, $37.0\ ^\circ\text{C}$ , to kelvin; (b) the boiling point of liquid nitrogen, $77\ \text{K}$ , to degrees Celsius; (c) a temperature rise of $15\ ^\circ\text{C}$ to kelvin.

Solution

(a) $T = 37.0 + 273.15 = 310.15\ \text{K} \approx 310\ \text{K}$ .

(b) $\theta = 77 - 273.15 = -196\ ^\circ\text{C}$ .

A platinum resistance thermometer

A platinum resistance thermometer has resistance $100.0\ \Omega$ in melting ice and $138.5\ \Omega$ in steam at standard pressure. In a liquid its resistance is $120.0\ \Omega$ . Calculate the temperature of the liquid on the scale of this thermometer, and explain why this may differ from the thermodynamic temperature.

Solution

Assuming resistance varies linearly with temperature between the fixed points:

\theta = \frac{120.0 - 100.0}{138.5 - 100.0} \times 100 = \frac{20.0}{38.5} \times 100 = 51.9\ ^\circ\text{C}

The resistance of platinum does not vary exactly linearly with thermodynamic temperature, so the empirical scale based on this assumption agrees with the thermodynamic scale only at the two fixed points.

Extrapolating to absolute zero

A fixed mass of gas is kept at constant volume. Its pressure is $1.000 \times 10^{5}\ \text{Pa}$ at $0\ ^\circ\text{C}$ and $1.366 \times 10^{5}\ \text{Pa}$ at $100\ ^\circ\text{C}$ . Assuming the pressure varies linearly with temperature, find the temperature in degrees Celsius at which the pressure would be zero.

Solution

Gradient: $\dfrac{(1.366 - 1.000) \times 10^{5}}{100} = 366\ \text{Pa K}^{-1}$ .

Pressure falls to zero after a further drop of $\dfrac{1.000 \times 10^{5}}{366} = 273\ \text{K}$ below $0\ ^\circ\text{C}$ :

\theta = -273\ ^\circ\text{C}

Every gas at low pressure gives the same intercept, independent of which gas is used. This is strong evidence that absolute zero is a property of temperature itself, not of any particular substance.

Two thermometers disagree

A thermocouple gives an e.m.f. of $0$ at $0\ ^\circ\text{C}$ and $4.10\ \text{mV}$ at $100\ ^\circ\text{C}$ . Placed in a beaker of water whose temperature is measured as $50.0\ ^\circ\text{C}$ by a gas thermometer, it gives $2.00\ \text{mV}$ . Calculate the temperature indicated by the thermocouple on its own linear scale and explain the discrepancy.

Solution

\theta = \frac{2.00}{4.10} \times 100 = 48.8\ ^\circ\text{C}

The gas thermometer reads $50.0\ ^\circ\text{C}$ , essentially the thermodynamic temperature. The thermocouple e.m.f. does not vary linearly with temperature, so its empirical scale differs from the thermodynamic scale between the fixed points. In practice a thermocouple is used with a calibration curve (e.m.f. against temperature measured with a reference thermometer), not with a linear assumption.

Choosing a thermometer

Suggest, with reasons, a suitable thermometer to measure (a) the rapidly changing temperature of a small soldering-iron tip; (b) the temperature of a large water bath to $\pm 0.1\ ^\circ\text{C}$ over the range $20$ to $60\ ^\circ\text{C}$ .

Solution

(a) A thermocouple: its junction is very small, so it has a tiny heat capacity, responds quickly and measures the temperature at a point; it also works at high temperatures (several hundred degrees Celsius).

(b) A platinum resistance thermometer (or a calibrated thermistor, which is very sensitive over a narrow range): accurate and precise, with a fairly linear response over this range; response time is not important for a large bath.

Calibrating a thermocouple or thermistor

Aim: produce a calibration curve so that a thermocouple or thermistor can be used as a thermometer.

Apparatus: thermocouple (one junction in melting ice as the reference) connected to a millivoltmeter, or a thermistor connected to an ohmmeter; a beaker of water heated by an electric kettle or immersion heater; a stirrer; a reference liquid-in-glass or digital thermometer.

Method:

Place the sensor and the reference thermometer close together in the water so they are at the same temperature.
Heat the water slowly while stirring continuously so that the temperature is uniform.
At regular intervals (for example every $5\ ^\circ\text{C}$ ) record the reference temperature and the e.m.f. (or resistance) at the same instant.
Repeat while the water cools, and average the readings at each temperature.
Plot e.m.f. (or resistance) against temperature and draw a smooth curve of best fit: this is the calibration curve.

Uncertainties and improvements: the sensor and thermometer may not be at exactly the same temperature (stir; place them together); the thermometer responds more slowly than the thermocouple (heat slowly, or switch off the heater and read while it settles); parallax reading an analogue thermometer (use a digital thermometer). Safety: hot water can scald; use a stable stand and keep electrical equipment away from water.

Exam tip

"State what is meant by thermal equilibrium": there is no net transfer of thermal energy between the regions, because they are at the same temperature.
"State two physical properties that vary with temperature and could be used to measure it": any two of the named examples. Give the property, not the thermometer ("resistance of a metal", not "a resistance thermometer").
"Explain why the thermodynamic scale is called absolute": it does not depend on the property of any particular substance, and it has a natural zero at the lowest possible temperature.
Questions on thermometers disagreeing want the idea that the property does not vary linearly with temperature, and that the scales agree only at the fixed points.

Summary

Thermal energy flows from higher to lower temperature.
Regions at the same temperature are in thermal equilibrium: no net energy transfer.
Any property that varies with temperature can be used for a thermometer: density of a liquid, volume of a gas at constant pressure, resistance of a metal, e.m.f. of a thermocouple.
Empirical scales assume linearity between fixed points, so different thermometers disagree between those points.
The thermodynamic scale does not depend on any substance; its zero, $0\ \text{K}$ , is absolute zero, the lowest possible temperature.
$T/\text{K} = \theta/^\circ\text{C} + 273.15$ ; temperature differences are the same in K and $^\circ$ C.

Practice questions

Question

Explain what is meant by two objects being in thermal equilibrium.
Convert $-40\ ^\circ\text{C}$ , $250\ ^\circ\text{C}$ and $4.2\ \text{K}$ to the other scale.
State four physical properties that vary with temperature and could be used to measure temperature.
A liquid-in-glass thermometer has a liquid column of length $22\ \text{mm}$ in melting ice and $182\ \text{mm}$ in steam. Calculate the temperature when the length is $96\ \text{mm}$ .
A resistance thermometer reads $25.0\ ^\circ\text{C}$ on its own scale when a gas thermometer reads $25.6\ ^\circ\text{C}$ . Explain how the two thermometers can be giving correct readings for their own scales but disagree.
State one advantage of a thermocouple over a liquid-in-glass thermometer when measuring the temperature of a small object whose temperature is changing quickly. Explain your answer.
Explain what is meant by absolute zero and why the thermodynamic scale is said not to depend on the property of any substance.
A thermistor has resistance $R$ given by $R = R_0 e^{b/T}$ , where $T$ is thermodynamic temperature, $R_0 = 0.020\ \Omega$ and $b = 3000\ \text{K}$ . (a) Calculate $R$ at $20\ ^\circ\text{C}$ . (b) Calculate the temperature, in $^\circ$ C, at which $R = 500\ \Omega$ . (c) Explain why a thermistor scale based on two fixed points would disagree with the thermodynamic scale.

Answers

They are at the same temperature, so there is no net transfer of thermal energy between them when in thermal contact.
$-40\ ^\circ\text{C} = 233\ \text{K}$ ; $250\ ^\circ\text{C} = 523\ \text{K}$ ; $4.2\ \text{K} = -269\ ^\circ\text{C}$ .
Density (volume) of a liquid; volume of a gas at constant pressure (or pressure at constant volume); resistance of a metal; e.m.f. of a thermocouple. (Resistance of a thermistor is also acceptable.)
$\theta = \dfrac{96 - 22}{182 - 22} \times 100 = \dfrac{74}{160} \times 100 = 46\ ^\circ\text{C}$ .
Each scale assumes its property varies linearly with temperature between the fixed points. Resistance and gas volume do not vary with temperature in exactly the same way, so the scales agree only at the fixed points and differ slightly in between.
The thermocouple junction is very small, so it has a small heat capacity: it reaches thermal equilibrium quickly (fast response) and measures the temperature at a point.
Absolute zero ( $0\ \text{K}$ ) is the lowest possible temperature, where internal energy is a minimum. The thermodynamic scale is defined theoretically (from absolute zero and the triple point of water), not by assuming that any material property varies linearly, so it does not depend on any substance.
(a) $T = 293.15\ \text{K}$ : $R = 0.020 \times e^{3000/293.15} = 0.020 \times e^{10.234} = 557\ \Omega$ . (b) $500 = 0.020e^{3000/T} \Rightarrow 3000/T = \ln(25\,000) = 10.127 \Rightarrow T = 296.2\ \text{K} = 23.1\ ^\circ\text{C}$ . (c) $R$ varies exponentially, not linearly, with $T$ , so a linear scale between two fixed points agrees with the thermodynamic scale only at those points.

Temperature and the direction of energy flow

Why thermometers work

Thermometric properties

Empirical scales and why thermometers disagree

The thermodynamic temperature scale

Absolute zero

Worked examples

Practice questions

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