Physics · Mechanics · Grade 7

Atmospheric pressure

Why does water boil on Mount Chimgan before reaching 100 °C?

The situation

You have gone from Tashkent to Chimyon for a day out. If you boil tea in the mountains, the water boils sooner, but the thermometer shows about 95 °C, not 100 °C — and meat and rice cook slowly. The bag of crisps you bought in Tashkent has puffed up and is as tight as a drum. What happens to the air as you go higher, and how does it change the boiling point?

Topic

Atmospheric pressure. Why water boils below 100 °C on Mount Chimgan

Key ideas

Atmospheric pressure
Boiling point
Torricelli's experiment

Formulas

p = p_{0} {(1 - \frac{L h}{T_{0}})}^{g M / (R L)}

The barometric formula (International Standard Atmosphere): $p_{0} = 101 325$ Pa, $T_{0} = 288.15$ K, $L = 0.0065$ K/m, exponent $g M / (R L) \approx 5.26$ . If the temperature is constant ( $L \to 0$ ) we get the isothermal formula $p = p_{0} e^{- M g h / (R T_{0})}$ .

p = ρ_{H g} g h_{H g}

Torricelli's barometer: the pressure of the mercury column equals the air pressure ( $ρ_{H g} = 13 595$ kg/m³).

\lg p = A - \frac{B}{C + T_{q}}

The boiling condition: the saturated vapour pressure of water becomes equal to the external pressure (the Antoine equation; $p$ in mmHg, $A = 8.071$ , $B = 1730.6$ , $C = 233.4$ ).

Q = m c (T_{q} - T_{0}), t = \frac{Q}{P}

The quantity of heat and the time needed to heat water to its boiling point ( $c = 4186$ J/(kg·K)).

p_{1} V_{1} = p_{2} V_{2}

Boyle's law: at constant temperature, when the pressure of the gas in a sealed bag falls, its volume increases.

Theory

At the bottom of an ocean of air

We live at the bottom of a layer of air hundreds of kilometres thick. Every square metre is pressed by the weight of the column of air above it: at sea level this is about 101 kPa — as if a load of 10 tonnes were placed on 1 m². The higher we climb, the less air is above us and the thinner it becomes, so the pressure falls: near the ground by about 1.2 kPa (≈ 9 mmHg) for every 100 m. This is exactly what the $p (h)$ graph on the panel shows. Air pressure can also be demonstrated at home: if you cover a glass full of water with paper and turn it upside down, the water does not spill; a syringe and a rubber suction cup also work thanks to the pressure of the outside air. In the 17th century Otto von Guericke had two metal hemispheres with the air pumped out (the Magdeburg hemispheres) pulled from both sides by dozens of horses (in 1654, 15 on each side) — the horses could not separate them. The bag of crisps in the scene is the same kind of demonstration: if the outside pressure falls, the bag swells.

Why does water boil at a lower temperature in the mountains?

For vapour bubbles to grow inside water, the saturated vapour pressure in them must overcome the external pressure. Vapour pressure grows quickly with temperature: 70 kPa at 90 °C, 101 kPa at 100 °C. In the mountains, where the pressure is lower, this condition is met at a lower temperature: in Chimyon water boils at ≈ 95 °C, on the summit of Everest at ≈ 70 °C. That is why food cooks slowly in the mountains. A pressure cooker, on the other hand, raises the pressure and so raises the boiling point to about 120 °C.

Torricelli's experiment and barometers

A tube closed at one end and nearly 1 m long is filled with mercury and inverted into a dish of mercury. The mercury falls a little and a vacuum (the Torricellian vacuum) is left at the top of the tube. The column is held up by the air pressing on the surface of the mercury in the dish: $p = ρ g h$ . At sea level $h = 760$ mm — this is the normal atmospheric pressure. Mercury is 13.6 times denser than water — with water this experiment would need a tube longer than 10 m. At home and in aircraft a liquid-free aneroid barometer is used: a thin corrugated metal capsule with the air pumped out is squeezed when the pressure rises and expands when it falls, and this deformation is passed to the needle through levers. An aneroid with a scale in metres is an altimeter. Manometers, which measure pressure in vessels, are in the “Pascal's law and hydraulics” experiment.

The barometric formula and the Boltzmann distribution

For a thin layer of air of thickness $d h$ , $d p = - ρ g d h$ , and for an ideal gas $ρ = p M / (R T)$ . If we take the temperature as constant, integration gives the isothermal barometric formula: $p = p_{0} e^{- M g h / (R T)}$ ; at $T = 288$ K the pressure falls by a factor of $e$ every ≈ 8.4 km. In the language of molecules this is the Boltzmann distribution: $n = n_{0} e^{- m_{0} g h / (k T)}$ — where the potential energy $m_{0} g h$ is large, there are few molecules. The dashed line on the $p (h)$ graph is the isothermal formula; in the real troposphere the temperature falls by about 6.5 °C per kilometre, cold air is denser, and so in the standard atmosphere (solid line) the pressure falls faster. The Maxwell distribution of molecular speeds is in the “Maxwell–Boltzmann distribution” experiment.

Model simplifications

The pressure is calculated from the International Standard Atmosphere model; in reality it varies by a few kPa with the weather (the pressure measured on the summit of Everest is ≈ 33–34 kPa, the model gives 31.4 kPa). The stove power $P$ is the useful power that reaches the water; heat losses are ignored, and the water is heated from 20 °C. The gas temperature in the bag is taken as constant, and the bag seam withstands an excess pressure of about 20 kPa. Time is speeded up 20 times, and the distant landscape is scaled down 10 times.

Did you know?

In 1643 the Italian scientist Evangelista Torricelli was the first to measure atmospheric pressure, using a glass tube of mercury. Normal atmospheric pressure: 101 325 Pa = 760 mm of mercury.
In 1648, at Blaise Pascal's request, his brother-in-law Florin Périer carried a barometer up the Puy de Dôme in France: on the summit the mercury column was markedly lower than at the foot. This experiment proved that air pressure decreases with height.

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.

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