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Lecture 21
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
• GayLussac's law
• The Combined Gas law
• Avogadro's number
• 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. 
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^{ o}C (blue data), 127^{ o}C (green data),
227^{ o}C (orange data), 327^{ o}C (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. P_{1}V_{1} = P_{2}V_{2} 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 20^{0}C the pressure of the gas when it occupies 3 litres is 100 kPa.
Find the pressure when the gas occupies 1.5 litres at 20^{0}C.
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. 
Absolute temperature
A temperature of 20^{ o}C is not twice as hot as 10^{ o}C 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^{ o}C, 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.
The Absolute temperature gives a proper zero to temperature.
To a greater accuracy, Absolute Zero is 273.15^{0}C so:
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
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^{ o}C or 273.16K.
Example
A sample of gas occupies 300ml at 27^{0}C and atmospheric pressure.
It is heated to 127^{0}C under constant pressure.
Find the new volume of the gas.
From Charles' law
Joe Louis GayLussac

GayLussac found and published Charles' notes on expansion with temperature along with his own experimental results,
so Charles' law is sometimes called GayLussac and Charles' law. 
The following is called GayLussac'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 27^{0}C and atmospheric pressure.
It is heated to 327^{0}C while the volume remains the same.
Find the new pressure of the gas.
From GayLussac's law
The Combined Gas Equation
All three gas laws can be combined to give one gas equation.
Example
At 73^{0}C, 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 127^{0}C and expanded to a volume of 0.1m^{3}.
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 GayLussac's law.
A section along the VT axis gives a similar straight line for Charles' law
Avogadro's Number
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 (N_{A}) for 1 gram mole of molecules is 6.023×10^{23}.
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 4157cm^{3} 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 273^{0}C.
GayLussac's law: For a given gas at constant volume, the ratio of pressure to Absolute temperature is constant.
Combined Gas equation:
Avogadro's number (N_{A}) for 1 g mol of molecules is 6.023×10^{23}
Universal Gas Equation: PV = nRT where n is the number of gram moles and R is the Universal Gas Constant.
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