Peter's Physics Pages

An Introductory Physics Course with Peter Eyland
Lecture 20 (the Gas Laws)

In this lecture the following are introduced:
• Boyle's law
• Charles' law and Absolute Temperature
• Gay-Lussac's law
• The Combined Gas law
• The Universal Gas Equation

Boyle's law Robert Boyle b. Jan. 25, 1627; d. Dec. 30, 1691 Robert Boyle was born in Lismore Castle in Ireland and went to school at Eton in England when he was 8. From 1641 to 1644 he toured Europe. He returned and settled in England, because of the troubles in Ireland. (These were the times of Charles, Cromwell and Owen Roe O’Neill). He joined a group, whose aim was to cultivate ideas called the "new philosophy." The new philosophy included new methods of experimental science, in which scientists sought to prove or disprove hypotheses through careful experiments. Boyle was the first person to isolate and collect a gas. He discovered "Boyle's law" in 1662. In the rest of Europe, this law is attributed to Edme Mariotte, so those countries tend to call this law by his name. (Mariotte did not publish his work until 1676.)

Boyle's apparatus Boyle used a "J" tube, sealed on the short "bulb" side and open on the long side. By slowly adding Mercury into the open side, the pressure in the bulb will be increased and the temperature kept constant.

Typical Results at 27 oC (blue data), 127 oC (green data), 227 oC (orange data), 327 oC (red data). The different coloured curves are for each separate temperature. Curves of equal temperature are called "isothermals". The graphs show that the product of pressure and volume is the same for each temperature. For example, Blue curve: 2000×1.5=1000×3, orange curve: 2500×2=1250×4.

Boyle's law

"For a given gas at a fixed temperature, the product of pressure and volume is a constant."

i.e. P1V1 = P2V2 when T is constant.
This is true for gases at low pressures only, and under such conditions most gases act as "ideal gases".

Example
A sample of air is confined to a cylinder fitted with a piston that can be moved to change the volume occupied by the gas. At 200C the pressure of the gas when it occupies 3 litres is 100 kPa. Find the pressure when the gas occupies 1.5 litres at 200C.

From Boyle's law Example
A sample of gas initially confined under a pressure of 50kPa is put into a 20 litre container under a pressure of 450kPa at the same temperature. Find the initial volume of the gas.

From Boyle's law Jacques Charles

Charles (1746 - 1823) investigated the effect of temperature on a fixed mass of gas while keeping the pressure constant. The straight lines mean that (keeping the pressure constant) the volume is proportional to the temperature. The lines meet at a temperature of -273 oC. Since you cannot have a negative volume, this is the lowest possible temperature. Absolute temperature

A temperature of 20 oC is not twice as hot as 10 oC because the Celsius scale has a zero arbitrarily set at the point where ice either melts or freezes. Charles' work enabled an Absolute temperature scale to be defined. Since the lowest possible temperature is -273 oC, this becomes the zero temperature on the Absolute scale. The unit of absolute temperature is called the Kelvin (K) and is the same size as the Celsius scale.

TKelvin = TCelsius + 273

The Absolute temperature gives a proper zero to temperature.

To a greater accuracy, Absolute Zero is -273.150C so:

TKelvin = TCelsius + 273.15

Charles' law can now be expressed this way:
"For a given mass of gas at constant pressure, the ratio of volume to absolute temperature is constant".
Most gases obey Charles' law at room temperatures and low pressures.

symbolically , for constant P

Charles' law computer demo

The Triple Point

To calibrate temperatures (i.e. to check a thermometer is reading correctly), a simple yet universal physical situation is needed. The triple point of water (where solid, liquid and vapour are in equilibrium at Atmospheric pressure) is such a situation. The triple point of water is 0.01 oC or 273.16K.

Example
A sample of gas occupies 300ml at 270C and atmospheric pressure. It is heated to 1270C under constant pressure. Find the new volume of the gas.

From Charles' law Joe Louis Gay-Lussac Gay-Lussac found and published Charles' notes on expansion with temperature along with his own experimental results, so Charles' law is sometimes called Gay-Lussac and Charles' law. Go here for detailed info on Gay-Lussac

The following is called Gay-Lussac's law.
"For a given gas at constant volume, the ratio of pressure to Absolute temperature is constant"

i.e. , for constant V

Example
A sample of gas occupies 500ml at 270C and atmospheric pressure. It is heated to 3270C while the volume remains the same. Find the new pressure of the gas.

From Gay-Lussac's law The Combined Gas Equation

All three gas laws can be combined to give one gas equation. Example
At -730C, a sample of gas occupies a volume of 0.1 litre under a pressure of 76kPa. Find the pressure of the gas when it is heated to 1270C and expanded to a volume of 0.1m3. Equation of State

The combined gas equation is called an "equation of state". This is because the three parameters (Volume, Temperature and Pressure) uniquely define the situation that the gas is in. As the state of the gas changes, the (V,T,P) point representing its state moves over a 3D surface.
Here is the surface for  A section along the PV axis gives Boyle's law. A section along the PT axis gives Gay-Lussac's law. A section along the VT axis gives a similar straight line for Charles' law

If the mass of the gas is increased by 50% i.e. The new surface of state lies a distance above the original surface. Now the mass of gas depends on the number of molecules present (more molecules - more gas). To measure the number of molecules, a unit is used called the gram mole. The gram mole is a number like a dozen (12), or score (20), or gross (144). Avogadro's number (NA) for 1 gram mole of molecules is 6.023×1023.

The Molar mass is the mass of Avogadro's number of molecules.

Eggsample :)
The mean radius of the Earth is 6370km. An egg has a volume of 50ml. For Avogadro's number of these eggs, find how high they would be stacked on the surface of the Earth. (Assume a smooth Earth's surface and perfect stacking with no spaces between eggs, so this will be a lower estimate). The Universal Gas Equation.

For 1 g mol of any gas, ,

or PV = RT where R is the universal Gas constant of 8.314 /(g mole.K).

For n g mol of gas, where ,

or PV = nRT where n is the number of gram moles and R is the Universal Gas Constant.
This is called the Universal Gas Equation.

Example
Find the volume occupied by 1 g mol of an ideal gas at Standard Temperature and Pressure (STP is 273K and 101.3kPa). Example
A sample of gas occupies a volume of 4157cm3 at 500K and 50kPa. Find
(a) the number of moles, and
(b) the number of molecules.  Example
The vacuum of interstellar space has about one molecule per cubic centimetre. Find the gas pressure if the temperture in the region averages 2K. Summarising:

Boyle's law: For a given gas at a fixed temperature, the product of pressure and volume is a constant.
Charles' law: For a given gas at constant pressure, the ratio of volume to absolute temperature is constant.
The unit of Absolute temperature is called the Kelvin (K) and is the same size as the Celsius scale. The Absolute temperature scale has a zero at -2730C.
Gay-Lussac's law: For a given gas at constant volume, the ratio of pressure to Absolute temperature is constant.
Combined Gas equation: Avogadro's number (NA) for 1 g mol of molecules is 6.023×1023

Universal Gas Equation: PV = nRT where n is the number of gram moles and R is the Universal Gas Constant. email Write me a note if you found this useful