Heat & Temperature

Rub your hands together and they warm up. Brake a bicycle and the brake pads heat. Where does the lost motion go, and what exactly is it that flows from a hot cup of tea into your fingers?

Hot bodies have more internal energy#

Whenever mechanical energy seems to disappear (friction, a crash, a sliding block slowing down), the bodies involved get warmer. The energy hasn’t vanished: it has gone into the internal energy of the bodies, the energy of their jiggling atoms and molecules. A hot body has more internal energy than an otherwise identical cold one.

Heat#

Put a hot body against a cold one: the hot one cools and the cold one warms. Internal energy passes from one to the other. Nothing is pushed through a distance, so no mechanical work is done. That makes it different from lifting a ball or a spring pushing a block.

Thermal equilibrium and the zeroth law#

Two bodies are in thermal equilibrium if no heat flows between them when they are put in contact. Experience shows:

The zeroth law lets us define temperature: bodies in thermal equilibrium have equal temperatures, and a hotter body has a higher temperature. Heat always flows from higher to lower temperature.

Walls that block or pass heat#

Real materials lie in between: good insulators (wool, foam, the vacuum of a flask) come close to adiabatic; metals are nearly diathermic.

The zeroth law tells us when two things are equally hot. To say how hot, we need a scale: a recipe for turning some measurable property into a number.

Building a scale#

Pick a substance and a property of it that changes steadily as it gets hotter, such as the length l of a mercury column in a glass capillary. Then:

1. Choose two easily reproduced fixed points: melting ice at 1 atm (the ice point) and boiling water at 1 atm (the steam point).
2. Assign them temperatures, and assume temperature is a linear function of the property: t = al + b.

The centigrade scale puts the ice point at 0° and the steam point at 100°. With l₀ and l₁₀₀ the column lengths there:

Fahrenheit

The Fahrenheit scale puts the ice point at 32°F and the steam point at 212°F, so 180 Fahrenheit degrees span the same interval as 100 Celsius degrees:

The platinum resistance thermometer#

The electrical resistance of a metal wire also rises steadily with temperature. Define the platinum scale by

A fine platinum coil wound on mica sits in a glass tube dipped into the bath. Its resistance is measured with a Wheatstone bridge: four resistances P, Q, R and X in a loop, with a galvanometer across the middle. There is no deflection (the bridge is balanced) when $P/Q = R/X$.

The leads to the platinum coil also have resistance, which changes with temperature too. A compensating wire of the same kind runs alongside them into the tube and is placed in the opposite arm, so the leads cancel. With P = Q and a sliding contact at distance x along a uniform bridge wire, the balance point moves linearly with R_t, and

Different scales disagree#

By definition the platinum scale makes resistance rise uniformly with its own temperature, and the mercury scale makes mercury expand uniformly with its own. Compare them and they agree at 0 and 100 (they were forced to) but not in between: measured on the mercury scale, platinum’s resistance rises slowly at low temperatures and faster at high ones. Each scale depends on its substance.

Mercury and platinum disagree about temperature. Gases, especially thin ones, almost all agree, and their agreement gives us the kelvin scale.

The constant-volume gas thermometer#

A gas is sealed in a bulb A connected by a capillary to a mercury manometer. The manometer’s other arm is open to the air and joined by a rubber tube to a movable mercury reservoir. Raising or lowering the reservoir keeps the mercury in the near arm at a fixed mark C, so the gas volume stays constant. The gas pressure is then

where p₀ is atmospheric pressure and h the difference in mercury levels. At constant volume, p rises steadily as the gas warms.

Defining T

Take the temperature to be proportional to the pressure, T = cp, and fix c with one reference: the triple point of water (ice, water and vapour together in equilibrium) is assigned 273.16 K. If the gas pressure there is p_tr:

A centigrade scale can be built the same way, from the pressures at the ice and steam points:

The ideal-gas scale#

Gas thermometers using different gases, or different amounts of the same gas, give slightly different readings for the steam point. But as the amount of gas (measured by p_tr) is made smaller, they converge to the same value, 373.15 K. So we define

This scale doesn’t depend on which gas is used. It turns out to be identical to the absolute temperature scale, which can be defined without reference to any substance at all.

The Celsius scale#

On the ideal-gas scale the ice point is 273.15 K and the steam point 373.15 K. The Celsius scale uses degrees of exactly the same size, shifted so the ice point is 0°C:

The old mercury centigrade scale is very close to it.

The ideal gas equation#

At low pressures (and far above their condensing temperatures) gases behave alike and obey

Callendar’s compensated thermometer#

Callendar removed the capillary error with a twin design. Bulb A (in the bath) and its capillary connect to a mercury-filled, graduated bulb B. An identical bulb C and capillary, holding an equal amount of the same gas, connect to the other side of a manometer. B and C sit in melting ice. When A is heated, mercury is drawn off from B until the manometer levels are equal again. Equating the amounts of gas on the two sides (the capillary terms cancel):

Railway lines have small gaps between rails, bridges sit on rollers, and a tight jar lid loosens under hot water. Almost everything grows a little as it warms.

Linear expansion#

A rod of length L at temperature T grows by ΔL when warmed by ΔT. The coefficient of linear expansion is

If α is small and nearly constant, with L₀ the length at 0°C and θ the Celsius temperature:

Volume expansion#

The odd behaviour of water#

Heat 1 g of water from 0°C: its volume falls, from about 1.00013 cm³ to 1.00000 cm³ at 4°C, and only then rises. Between 0 and 4°C its expansion coefficient is negative, so water is densest at 4°C.

In winter, lake water at 4°C sinks to the bottom and stays there, while colder water and then ice float on top. The ice insulates the water below, and fish survive near the bottom.

Real and apparent expansion of a liquid#

Heat a liquid in a flask with a graduated stem. At first the flask (heated from outside) expands before the liquid does, and the level drops, as though the liquid had shrunk. Then the liquid catches up and rises. What you read on the stem is the apparent expansion: