Physics · Molecular physics and thermodynamics · Grade 9

Gas under a piston

Why does tyre pressure drop in winter, and why does a pump get hot?

The situation

On a cold winter morning the pressure in a car tyre drops, and in summer it rises. When you pump up a bicycle tyre, the pump gets warm, and in a diesel engine the air is compressed so much that the fuel ignites by itself without a spark. The cause of it all is the same: a gas is billions of molecules constantly striking the walls. Watch the nitrogen molecules under the piston, slowed down 2000 times.

Topic

Ideal gas and the molecular-kinetic theory. Bicycle pump, tyre and diesel engine

Key ideas

Ideal gas equation of state
Molecular-kinetic theory
Adiabatic process

Formulas

p V = ν R T, k = \frac{R}{N_{A}}

The equation of state of an ideal gas (Mendeleev–Clapeyron). $ν$ is the amount of substance, mol; $k$ is the Boltzmann constant.

p = \frac{2}{3} n_{0} ⟨ E_{k} ⟩, ⟨ E_{k} ⟩ = \frac{3}{2} k T

The basic equation of the kinetic theory: the pressure from the concentration of molecules and their mean kinetic energy; temperature is the measure of this energy.

Q = Δ U + A^{'}, Δ U = \frac{i}{2} ν R Δ T, A^{'} = \int p d V

The first law of thermodynamics: the heat given to a gas increases its internal energy and the gas does work ( $i$ is the number of degrees of freedom).

C_{p} - C_{V} = R, p V^{γ} = const, γ = \frac{C_{p}}{C_{V}} = \frac{i + 2}{i}

Mayer’s relation and the adiabat (Poisson’s equation). He: $γ = 5 / 3$ , N₂: $7 / 5$ , CH₄: $4 / 3$ .

f (v) = 4 π {(\frac{m_{0}}{2 π k T})}^{3 / 2} v^{2} e^{- m_{0} v^{2} / 2 k T}

The Maxwell distribution: how the molecules are distributed by speed.

Theory

Pressure is the sum of impacts

Each molecule that hits a wall and bounces back gives it a momentum of $2 m_{0} v_{n}$ . The average of trillions of impacts per second is the pressure we measure. The calculation gives the basic equation of the kinetic theory: $p = \frac{1}{3} n_{0} m_{0} ⟨ v^{2} ⟩ = \frac{2}{3} n_{0} ⟨ E_{k} ⟩$ (the virial theorem gives the same result: $p V = \frac{2}{3} E_{t r}$ ). The simulation measures the pressure in exactly this way — by counting the momentum given to the wall and the piston — and compares it with $p V = ν R T$ . In a mixture of gases each gas makes its own impacts, and the pressures add: $p = p_{1} + p_{2} + \dots$ (Dalton’s law). There are only 400 “representative” molecules in the scene, so the needle of the pressure gauge trembles: the relative fluctuation is $\sim 1 / \sqrt{N}$ (in detail — in the “Phase transition and fluctuations” experiment); with a real $10^{22}$ molecules it is not noticeable.

Temperature, the Boltzmann constant and speeds

When a gas comes into thermal equilibrium with the walls, the temperatures become equal — the thermometer is based on this. Temperature is a measure of the mean energy of the translational motion of the molecules: $⟨ E_{k} ⟩ = \frac{3}{2} k T$ . The Boltzmann constant $k = R / N_{A} = 1.38 \cdot 10^{- 23}$ J/K links the energy of one molecule with the kelvin, and the universal gas constant $R = 8.31$ J/(mol·K) is the same relation for 1 mol. The absolute scale: $T = t + 273.15$ . The isobaric and isochoric straight lines are extrapolated to $t = - 273.15$ °C (see it in the log) — real gases turn into a liquid before that. Speeds are not all the same — the Maxwell distribution: the most probable $v_{p} = \sqrt{2 k T / m_{0}}$ , the mean $⟨ v ⟩ = \sqrt{8 k T / π m_{0}}$ , the root-mean-square $v_{r m s} = \sqrt{3 k T / m_{0}}$ ; for nitrogen at 300 K ≈ 422, 476 and 517 m/s (the panel shows values measured from the molecules). Measuring speeds experimentally (the Stern experiment) — in the “Maxwell–Boltzmann distribution” experiment.

The four processes and the gas laws

Isothermal ( $T$ = const): the walls hold the temperature, $p V$ = const (Boyle–Mariotte law). Isobaric ( $p$ = const): the load on the piston holds the pressure, the heated gas expands, $V / T$ = const (Gay-Lussac’s law), $V = V_{0} (1 + α t)$ , $α = 1 / 273.15$ K⁻¹. Isochoric ( $V$ = const): the piston is fixed, $p / T$ = const (Charles’s law), $p = p_{0} (1 + β t)$ ; the thermal pressure coefficient $β$ is also $1 / 273.15$ K⁻¹ — this is why a tyre goes “soft” in winter. Adiabatic ( $Q = 0$ ): the gas heats up when compressed and cools when it expands. All of them are special cases of the polytropic process: $p V^{n}$ = const, with $n = 1; 0; \infty; γ$ respectively. In engineering: in a diesel engine adiabatic compression ignites the fuel, pneumatic brakes and hammers work on compressed air, and heating a gas cylinder is dangerous — in an isochoric process the pressure rises in proportion to the temperature.

Internal energy, work and heat

The internal energy of an ideal gas is the sum of the kinetic energies of its molecules: $U = \frac{i}{2} ν R T$ . It can be changed in two ways: by doing work (a moving piston speeds up or slows down a molecule that hits it) and by transferring heat (a hot wall sends a molecule back faster). The first law: $Q = Δ U + A^{'}$ ; the work done by the gas is $A^{'} = \int p d V$ — the area under the curve in the p–V diagram, and in an isothermal process $A^{'} = ν R T \ln (V_{2} / V_{1})$ . In the isoprocesses: isothermal — $Δ U = 0$ , $Q = A^{'}$ ; isochoric — $A^{'} = 0$ , $Q = Δ U$ ; isobaric — $Q = Δ U + p Δ V$ ; adiabatic — $Q = 0$ , $A^{'} = - Δ U$ . The “First law” section of the panel counts $Q$ , $A^{'}$ and $Δ U$ from the energy of each molecule. The units of work and heat were linked by Joule’s experiment: the mechanical equivalent of heat, 1 cal ≈ 4.19 J (in the “Calorimeter: heat balance” experiment).

Heat capacity, Mayer’s relation and γ

If we heat a gas at $V$ = const, all the heat goes into $U$ : the molar heat capacity is $C_{V} = \frac{i}{2} R$ . At $p$ = const the gas also expands and does work, so $C_{p} = C_{V} + R$ (Mayer’s relation), $γ = C_{p} / C_{V}$ . Each degree of freedom has on average an energy of $\frac{1}{2} k T$ (the equipartition theorem): for monatomic He $i = 3$ , for diatomic N₂ $i = 5$ (3 translational + 2 rotational), for polyatomic CH₄ $i = 6$ — in the panel, look at the share of translational energy $3 / i$ . The vibrations of the atoms are almost “frozen” at room temperature: this is explained by quantum theory (“Quantum statistics”); in real methane the vibrations contribute a little and $γ \approx 1.3$ . In an adiabatic process $p V^{γ}$ and $T V^{γ - 1}$ stay constant (Poisson’s equation); in a polytropic process $C_{n} = C_{V} (n - γ) / (n - 1)$ . Measuring $γ$ : the Clément–Désormes method (in the panel) — the gas expands adiabatically ( $p_{1} \to p_{0}$ ), then at constant volume it warms up to the wall temperature ( $p_{0} \to p_{2}$ ): $γ = \ln (p_{1} / p_{0}) / \ln (p_{1} / p_{2})$ . In Rüchardt’s method a small ball in the neck of a vessel oscillates on the gas “spring”: $ω^{2} = γ p S^{2} / (m V)$ .

The laws of thermodynamics and the equations of state

The zeroth law: two bodies that are each in equilibrium with a third body are also in equilibrium with each other — temperature is defined by this. The first law is the conservation of energy. The second and third laws (entropy, Nernst’s theorem) — in the “Entropy and the arrow of time” experiment. A gas is described by two equations: the thermal one, $p = p (V, T)$ , where $p V = ν R T$ , and the caloric one, $U = U (V, T)$ , where $U = \frac{i}{2} ν R T$ . They are linked: from Maxwell’s relation ${(\partial S / \partial V)}_{T} = {(\partial p / \partial T)}_{V}$ follows ${(\partial U / \partial V)}_{T} = T {(\partial p / \partial T)}_{V} - p$ , which for an ideal gas equals zero — $U$ does not depend on volume (Joule’s law). Check it: in adiabatic mode the “Free expansion” button pulls the piston away instantly — the gas does no work, receives no heat and its temperature does not change, whereas in a slow adiabatic expansion it cools. In general $C_{p} - C_{V} = T V α^{2} / ϰ_{T}$ ( $α$ is the volume expansion coefficient, $ϰ_{T}$ the isothermal compressibility); for an ideal gas this is $ν R$ . $U$ , $V$ , $C$ are proportional to the number of particles (extensive), while $p$ and $T$ are not (intensive). The cooling of a real gas on throttling (the Joule–Thomson process) — in the “Real gases” experiment.

Did you know?

In a diesel engine the air is compressed 15–20 times and heats to over 500 °C — the fuel ignites by itself when it is injected, no spark plug is needed.
At room temperature air molecules move at an average of ≈ 500 m/s — faster than sound — but each one collides with other molecules billions of times a second and changes direction.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Melting, boiling and evaporation — Grade 7
Calorimeter: heat balance — Grade 7
Tandoor and sandal: heat — Grade 7
Humidity and dew — Grade 9
Heat engine: the Otto cycle — Grade 9
Entropy and the arrow of time — Grade 9
Brownian motion and diffusion — Grade 9
Surface tension and capillarity — Grade 9
Crystalline and amorphous solids — Grade 9
Phase transitions — Grade 10