Temperature and its measurement
Heat and temperature
Pour hot tea into a steel tumbler on a cold winter morning. Touch it after a few minutes: the tumbler is warm, and the tea is a little cooler.
We describe this with temperature: a measure of how hot or cold a body is. The hot tea is at a higher temperature than the tumbler around it.
Watch energy flow from the tea into the tumbler, and from the tumbler into the cold air. Energy that flows because of a temperature difference is called heat.
Heat always flows from the higher temperature to the lower temperature, never the other way on its own. A glass of iced water warms up; a hot cup cools down.
The flow stops when both bodies reach the same temperature. They are then in thermal equilibrium.
So temperature and heat are not the same thing. Temperature tells us which way energy will flow. Heat is the energy that actually flows.
Heat is energy, so its SI unit is the joule. Temperature has its own SI unit, the kelvin. In daily life we also use degrees Celsius.
Compare a bucket of warm bath water with a spoonful of boiling water. The spoonful is at the higher temperature, but the bucket can give out far more heat.
Thermometers and temperature scales
To measure temperature we use a thermometer. It relies on some property of a substance that changes steadily with temperature.
The most common one uses the expansion of a liquid. Mercury or coloured alcohol rises up a thin glass tube as it gets warmer.
To make a scale we need two fixed points that are easy to reproduce. The usual choices are the ice point, where pure water freezes, and the steam point, where pure water boils, both at standard atmospheric pressure.
On the Celsius scale, the ice point is 0 degrees and the steam point is 100 degrees. The gap between them is split into 100 equal parts.
On the Fahrenheit scale, the same two points are marked 32 and 212. So the same gap is split into 180 parts.
Put the two thermometers side by side. Any temperature cuts both columns at the same fraction of the way from the ice point to the steam point.
That gives t F minus 32, over 180, equals t C over 100. Rearranged: t F equals nine fifths of t C, plus 32.
Try body temperature, 37 degrees Celsius. Nine fifths of 37 is 66.6. Add 32 and you get 98.6 degrees Fahrenheit, the number on a clinical thermometer.
And here is a curious one: at minus 40, both scales show the same number. Minus 40 degrees Celsius is also minus 40 degrees Fahrenheit.
The ideal-gas law and absolute zero
Liquid thermometers have a problem. Mercury and alcohol do not expand in exactly the same way, so away from the fixed points two such thermometers can disagree slightly.
Gas thermometers agree with each other far better, whichever gas is used, as long as the gas is at low density. Let us see why.
At low density, all gases obey the same simple laws. Boyle’s law: at fixed temperature, P V is constant. Charles’ law: at fixed pressure, V over T is constant.
Together they give the ideal-gas equation, P V equals μ R T. Here μ is the number of moles, and R, the universal gas constant, is 8.31 joules per mole per kelvin.
Now keep the volume of a gas fixed and heat it. Its pressure rises in a straight line with temperature. This is a constant-volume gas thermometer.
Plot pressure against temperature in degrees Celsius for three different gas samples. You get three straight lines with different slopes.
Extend the lines backwards. Every one of them reaches zero pressure at the same temperature: minus 273.15 degrees Celsius.
This is absolute zero, the lowest temperature possible. It is the zero of the absolute, or Kelvin, temperature scale.
One kelvin is the same size as one degree Celsius, so T equals t C plus 273.15. Ice melts at 273.15 kelvin, and water boils at 373.15 kelvin.
On this scale, the pressure of a gas at constant volume is simply proportional to T. Double the kelvin temperature and the pressure doubles.
Try it: three scales and a gas thermometer
- Drag the temperature slider and watch all three thermometers move together.
- Every gas line is straight, and every one reaches P = 0 at −273.15 °C.
- Try −40 °C, and try doubling the kelvin temperature of gas A.
Heat or temperature?
Heat flows from body A to body B when they are placed in contact. What must be true?
- A contains more heat than B
- A is at a higher temperature than B
- A has a larger mass than B
- A has a larger specific heat than B
Answer
B. Heat is energy in transit, so a body does not “contain heat”. It always flows from the higher temperature to the lower one, whatever the masses. A small red-hot nail dropped into a big bucket of water still gives heat to the water.
Fever reading
A thermometer shows a child’s temperature as 104 °F. What is it in degrees Celsius and in kelvin?
- 40 °C and 313.15 K
- 40 °C and 233.15 K
- 57.8 °C and 330.95 K
- 72 °C and 345.15 K
Answer
A. Use tC = (5/9)(tF − 32) = (5/9)(104 − 32) = (5/9)(72) = 40 °C. Then T = tC + 273.15 = 313.15 K. Forgetting to subtract 32 gives 57.8 °C; forgetting to multiply by 5/9 gives 72 °C.
Summary
- Celsius and Fahrenheit:
(t_F − 32)/180 = t_C/100, so t_F = (9/5)t_C + 32 - Kelvin:
T = t_C + 273.15 - Ideal-gas equation:
PV = μRT, R = 8.31 J mol⁻¹ K⁻¹ - Constant-volume gas:
P ∝ T
- Temperature measures hotness. Heat is energy that flows from a higher to a lower temperature until thermal equilibrium is reached.
- A scale needs two fixed points: ice point (0 °C, 32 °F) and steam point (100 °C, 212 °F).
- Gas thermometers at low density agree whatever the gas. Extrapolated, every P–t line reaches zero pressure at −273.15 °C, absolute zero.
- A kelvin is the same size as a degree Celsius; only the zero is shifted.