In this lecture the following are introduced
• Avogadro's number
• Brownian motion
• Boyle's, Charles' and Gay-Lussac's Laws
• The Ideal Gas Equation
• The Work Done by an Ideal Gas at Constant Temperature

Avogadro's Number

Amedeo Avogadro (1776 - 1856) proposed Avogadro's Principle (1811)
Equal volumes of different gases at the same temperature and pressure contain equal numbers of molecules.

Jean Perrin (1908) is the first person to have used the term Avogadro's number for the number of particles in a gram mole. Avogadro had no idea of moles or of the number that commemorates him. The gram mole is now the number of atoms in 12 grams of Carbon¹², i.e. 6.0221367 x 10²³. Carbon¹² was chosen arbitrarily to serve as the reference standard of the mole unit for the International System of Units (SI). Perrin estimated it to be between 6.5 x 10²³ and 6.9 x 10²³ from studies of Brownian motion.

Brownian motion

Brownian motion was first observed by Jan Ingenhousz in 1785, but was subsequently rediscovered by Robert Brown in 1828. Brown used a microscope to observe smoke particles in a light beam. He saw specks of light moving about erratically and apparently unpredictibly.

Albert Einstein (1879 - 1955) made an estimate of 6.03 × 1023 for Avogadro's number from Brownian motion.

Go here for computer demo of Brownian motion.
Modern methods of determining Avogadro's number rely on the use of x-ray crystallography to get precise dimensions in crystals.

Ideal Gases

For more information on Ideal Gases, go here

Boyle's Law (1662)
For a given gas at a fixed temperature, the product of pressure and volume is a constant.

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

Charles' Law (1787)

For a given gas at constant pressure, the ratio of volume to absolute temperature is constant.

Charles' Law 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

Gay-Lussac's Law (1802)
For a given gas at constant volume, the ratio of pressure to Absolute temperature is constant.

Gay-Lussac's Law 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

Combined Gas Law
All three of these gas laws can be combined to give one gas equation.

Combined Gas Law 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.

The Ideal Gas Equation
The conbined gas law can re-stated as an equation of state using the gram mole.

Where
R = 8.314 J. (g mole)^-1.K^-1 is the Universal Gas Constant
n is the number of gram moles

Ideal Gas Equation Example 1
Find the volume occupied by 1 g mol of an ideal gas at Standard Temperature and Pressure (STP is 273K and 101.3kPa).

Ideal Gas Equation Example 2
A sample of ideal gas occupies a volume of 4157cm³ at 500K and 50kPa. Find
(a) the number of moles, and
(b) the number of molecules.

Ideal Gas Equation Example 3
Find the number of molecules in 1 cm³ of any gas at standard temperature and pressure.

This is called the Loschmidt number (hence the "L" subscript). It is an absolute number in that it does not depend on the size of the mole like Avogadro's number.

Work done by an Ideal Gas at Constant Temperature

At constant temperature, the ideal gas equation gives Boyle's law:

On a p vs V diagram lines of equal temperature or isothermals, are shown in the diagram.

The work done by the system in an isothermal change is given by:

For an expansion, V_f > V_i where the logarithm is positive so the work done by the gas is positive when it expands.

Example
Two moles of Oxygen gas (assumed ideal) expand at a constant temperature of 310 K from 12 litres to 19 litres. Find the work done by the gas in expanding.

Summarising:

The gram mole is the number of atoms in 12 grams of Carbon¹².
A gram mole contains Avogadro's number of molecules, i.e. 6.02 x 10²³
Boyle' Law: , for constant Temperature (T).
Charles' Law: , for constant Pressure (p).
Gay-Lussac's Law: , for constant Volume (V).
Ideal Gas Equation:
Work done by an Ideal Gas at Constant Temperature (T):

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