10. Gases

Because mercury is about 13.5 times denser, its column is 13.5 times shorter: 76 cm instead of over 10 m. A water barometer needs a tube over ten metres tall. Mercury also has a very low vapour pressure, so the space above it is a good vacuum.

No. A wider column weighs proportionally more but also presses on a proportionally larger base, so the weight per area, which is the pressure, depends only on the height and the density. Tilt the tube and the height (vertical) stays the same while the column gets longer.

The air pressing down on the surface in the dish. That pressure is transmitted through the liquid to the bottom of the column, where it balances the column's weight. Take the tube to a mountain top and there is less air above you, so the column is shorter. This is how an altimeter works.

Because warming a gas makes it push harder, or expand. Boyle kept his glass tube and the room at a steady temperature. The next experiment lets the temperature vary and keeps the pressure fixed instead.

Boyle did not know, and nor did anyone for another century. A good modern answer: pressure comes from particles hitting the walls, and squeezing the same number of particles into half the space makes them hit twice as often. A later section of this unit builds exactly that from colliding particles and shows the law falling out.

No. At very high pressures, or low temperatures, gas molecules take up space and attract one another, and real gases depart from it. A later section puts a number on the departure for real gases.

No. Every real gas turns to a liquid and then a solid long before it gets there, and the line is only an extrapolation. Absolute zero also cannot be reached by any finite process. Helium comes closest: it stays a gas down to 4 K, and the coldest experiments in laboratories reach within billionths of a kelvin of it.

Because all gases, at low density, respond to temperature in the same way: their pressure or volume is proportional to the kelvin temperature, whatever the gas. That the zero does not depend on the gas was the first sign that there is something universal about temperature, and that the kelvin scale is not an accident of water.

Because the laws say V ∝ T, and a proportion only works from a true zero. Doubling the temperature from 10 °C to 20 °C does not double the volume; doubling it from 283 K to 566 K does. Using Celsius in a gas law can give a negative volume.

Because at ordinary pressures a gas is mostly empty space: the molecules are far apart and spend almost all their time flying freely, so their size and their attractions hardly matter. What matters is how many there are, how hot, and how much room they have. The last sections show why.

Not necessarily. R has a value for every set of units: 8.314 J/(mol K) in SI, 0.082057 L atm/(mol K) in litres and atmospheres, and 62.36 L mm Hg/(mol K) in the units of the old barometer. All describe the same thing; choose the one that matches your data and the units cancel on their own.

The molar mass of air is about 29 g/mol. Vapours of ethanol (46), acetone (58) and chloroform (119) are heavier, so at the same temperature and pressure the vapour is denser than air and sinks and pools along a floor or bench, which is why such vapours are a fire hazard at ground level.

In a smaller box the particles meet the walls more often, because they have less distance to cover between hits. Pressure is the rate at which momentum is handed to a wall, so the same particles at the same speeds push harder. Halve the volume and the hits are twice as frequent, which is Boyle's law.

Temperature measures the mean kinetic energy, ½ m v², and the same temperature means the same energy for every gas. For a heavier particle the same energy gives a lower speed: v ∝ 1/√m. In air at room temperature, hydrogen averages 1900 m/s, nitrogen 515 m/s.

Because a molecule collides some billions of times a second and goes only a tenth of a micrometre or so between collisions, so it zigzags. It races, but its path is a random walk, and random walks make slow progress. This slow spreading is diffusion, and it is why a smell takes seconds to cross a room even though the molecules move at hundreds of metres per second.

Because a measures the pull between molecules. Polar and large molecules attract one another more strongly (the forces of Unit 6), so a is larger and they condense at higher temperatures: that is the reason ammonia liquefies under pressure at room temperature and helium needs 4 K.

At low pressure and high temperature, where molecules are far apart and fast. At 1 atm and 25 °C every gas here has Z within 1 per cent of 1, which is why PV = nRT serves so well for ordinary work.

From the size of the disks, the same effect as b. The simulated gas was a little more than ideal (the ratio 1.03 or so), and the excess matched the excluded-area correction 2πr²N/A. Real gases have it too, along with the attractions, which the simulation left out.