Theory

Temperature, heat and the first law of thermodynamics

How temperature is measured, how heat changes the temperature and state of bodies, how an ideal gas is described and how the first law connects heat, work and internal energy in processes and cycles.

On this page

Temperature and thermal equilibrium
Thermal expansion

Temperature and thermal equilibrium

1. Thermodynamic system and state variables

A thermodynamic system is a portion of matter separated from the rest, the surroundings, by a real or ideal surface; a gas enclosed in a cylinder with a piston is the typical example. The system is closed if it exchanges no matter with the surroundings, isolated if it exchanges neither matter nor energy. The equilibrium state of a system is described by a few macroscopic quantities, the state variables: for a gas they are the pressure $p$ , the volume $V$ , the temperature $T$ and the amount of substance $n$ . A quantity that depends only on the state, and not on the way the system reached it, is called a state function; internal energy will be the main example.

2. Thermal equilibrium

Two systems placed in contact through a wall that allows exchanges of energy, called a diathermic wall, evolve until their state variables stop changing; at that point they are said to be in thermal equilibrium. A wall that prevents these exchanges is called adiabatic, and a thermos approximates a container with adiabatic walls. Every temperature measurement is based on thermal equilibrium, because the thermometer indicates its own state once it has reached equilibrium with the body being measured.

3. Zeroth law of thermodynamics

If two systems $A$ and $B$ are each in thermal equilibrium with a third system $C$ , then $A$ and $B$ are in thermal equilibrium with each other.

It is a law, that is, a statement derived from experience and not provable from other laws. It makes it possible to define temperature as the quantity that takes the same value in all systems in thermal equilibrium with one another, and it justifies the use of the thermometer. The thermometer plays the role of system $C$ ; if it reads the same value in contact with $A$ and with $B$ , then the two bodies are in thermal equilibrium even without ever having been placed in contact.

4. Temperature scales

A temperature scale assigns a number to temperature by means of a quantity that varies with it, such as the length of a column of mercury or the pressure of a gas at constant volume, and of two reproducible fixed points. Three scales are in use.

Celsius scale: $0^{\circ} C$ is the melting point of ice and $100^{\circ} C$ the boiling point of water at a pressure of $1 atm$ .
Kelvin, or absolute, scale, which is the one of the International System: it has the same intervals as the Celsius scale and its zero at absolute zero, the minimum temperature, which cannot be reached; $T = t_{C} + 273.15$ holds, with $T$ in kelvins and $t_{C}$ in degrees Celsius.
Fahrenheit scale: ice melts at $32^{\circ} F$ and water boils at $212^{\circ} F$ , therefore one hundred degrees Celsius correspond to one hundred eighty degrees Fahrenheit and $t_{F} = \frac{9}{5} t_{C} + 32$ .

Since the kelvin and the degree Celsius have the same size, a temperature difference has the same value on the two scales: $Δ T = 20 K$ is equivalent to $Δ t_{C} = 20^{\circ} C$ . In formulas that contain $T$ and not $Δ T$ , such as the equation of state of gases, the temperature in kelvins is used instead. For example $37^{\circ} C$ corresponds to $310.15 K$ and to $\frac{9}{5} \cdot 37 + 32 = 98.6^{\circ} F$ .

Thermal expansion

1. Linear expansion

When the temperature of a solid increases, its dimensions generally increase. For a rod of length $L_{0}$ at temperature $T_{0}$ , over a not too wide temperature interval, the elongation is proportional to the initial length and to the change in temperature:

Δ L = α L_{0} Δ T, L = L_{0} (1 + α Δ T) .

The constant $α$ is the coefficient of linear expansion; it is measured in $K^{- 1}$ and depends on the material; it is about $1.2 \cdot 1 0^{- 5} K^{- 1}$ for steel and $2.3 \cdot 1 0^{- 5} K^{- 1}$ for aluminum. The law is experimental and is used to size the expansion joints of bridges and railway tracks. A steel rail $10 m$ long that warms up by $30 K$ lengthens by $1.2 \cdot 1 0^{- 5} \cdot 10 \cdot 30 m = 3.6 mm$ .

2. Area and volume expansion

Let a homogeneous and isotropic solid have coefficient of linear expansion $α$ . For small changes in temperature, an area $S_{0}$ and a volume $V_{0}$ of the solid change by

Δ S = 2 α S_{0} Δ T, Δ V = β V_{0} Δ T with β = 3 α .

Indeed, for a cube of edge $ℓ_{0}$ one has $V = ℓ_{0}^{3} (1 + α Δ T)^{3} = V_{0} (1 + 3 α Δ T + 3 α^{2} Δ T^{2} + α^{3} Δ T^{3})$ , and since $α Δ T$ is of the order of $1 0^{- 3}$ the terms with $α^{2}$ and $α^{3}$ are negligible; for the area the reasoning is the same with the square. A hole in a plate expands as if it were filled with the same material, and the hot fitting of metal rings onto a shaft is based on this fact. For liquids only volume expansion is considered, with a coefficient $β$ measured directly and generally larger than that of solids. When a liquid fills a container to the brim and the two are heated together, the volume that overflows is $(β_{ℓ} - 3 α_{r}) V_{0} Δ T$ , where $β_{ℓ}$ is the coefficient of the liquid and $α_{r}$ that of the container.

Behavior of water

Between $0^{\circ} C$ and $4^{\circ} C$ water contracts when it is heated, therefore its density is maximum at about $4^{\circ} C$ . For this reason, in a lake that freezes in winter the water at the bottom stays at about $4^{\circ} C$ , while the ice, less dense than liquid water, forms at the surface.

The theory continues with 8 more sections (heat and calorimetry, changes of state and heat transfer, ideal gas and equation of state…), plus all worked examples, the solutions and the PDF.

Unlock now