Avogadro’s law/ Avogadro’s hypothesis states that under the same conditions of temperature and pressure, equal volumes of different gases contain an equal number of molecules. Under the assumption of a perfect or ideal gas, we can derive this empirical relation from the kinetic theory of gases. The law is approximately valid for real gases at sufficiently low pressures and high temperatures. The law is named after Amedeo Avogadro.

**Mathematical Definition**

Avogadro’s hypothesis is represented by the formula V ∝ n or V/n = k——-①

V = volume of the gas;

n = amount of substance of the gas (measured in moles);

k = constant for a given temperature and pressure.

This law describes how, under the same condition of temperature and pressure, equal volumes of all gases contain the same number of molecules. For comparing the same substance under two different sets of conditions, the law can be usefully expressed as follows:

The equation shows that, as the number of moles of gas increases, the volume of the gas also increases in proportion. Similarly, if the number of moles of gas is decreased, then the volume also decreases. Thus, the number of molecules or atoms in a specific volume of an ideal gas is independent of their size or the molar mass of the gas.

*Derivation Of Avogadro’s Hypothesis From The Ideal Gas Law*

Ideal gas law states that

PV = nRT

Where,

R = the gas constant,

T = the temperature (in Kelvin),

P = the pressure (in pascals).

Comparing equation (3) and equation (1) we get,

The above equation shows that RT/P is constant for a fixed temperature and pressure.

An equivalent formulation of the ideal gas law can be written using Boltzmann constant k_{B}, as PV=Nk_{B}T

Where, N = the number of particles in the gas,

and the ratio of R over k_{B} is equal to the Avogadro constant.

In this form, for V/N is a constant, we have

If T and P are taken at standard conditions for temperature and pressure (STP), then k′ = 1/n_{0}, where n_{0} is the Loschmidt constant.

**Applications Of Avogadro’s Hypothesis**

1. In explaining Gay Lussac’s law of gaseous volumes.

2. In determining the atomicity of gasses.

3. In determining the molecular formula of a gas

4. In establishing the relationship between relative molecular mass and vapour density.

*Significance Of Avogadro’s Hypothesis*

Discovering that the volume of a gas was directly proportional to the number of particles it contained was crucial in establishing the formulas for simple molecules at a time (around 1811) when the distinction between atoms and molecules was not clearly understood. In particular, the existence of diatomic molecules of elements such as H_{2}, O_{2}, and Cl_{2} was not recognized until the results of experiments involving gas volumes was interpreted.

Early chemists calculated the molecular weight of oxygen using the incorrect formula HO for water. This lead to the molecular weight of oxygen being miscalculated as 8, rather than 16. However, when chemists found that an assumed reaction of H + Cl → HCl yielded twice the volume of HCl, they realized hydrogen and chlorine were diatomic molecules. The chemists revised their reaction equation to be H_{2} + Cl_{2} → 2HCl.

When chemists revisited their water experiment and their hypothesis that

HO → H + O they discovered that the volume of hydrogen gas consumed was twice that of oxygen. By Avogadro’s Law, this meant that hydrogen and oxygen were combining in a 2:1 ratio. This discovery led to the correct molecular formula for water (H_{2}O) and the correct reaction 2H_{2}O → 2H_{2} + O_{2}

*Problem Faced By Avogadro Regarding His Hypothesis*

At his time many scientists (famous chemists like Gay Lussac) were not quite familiar with this kind of behaviour of gases. So they found it difficult to understand and accept Avogadro’s hypothesis.

To remedy this, Avogadro introduced the term called ‘molecules’ which can be referred to as the combinations of tiny particles. He also insisted on its origin, which is: the word molecule was derived from an old term ‘mole’ which means ‘lumps of matter’. Hence, a molecule can be viewed as a small cluster of matter.

*Historical Account And Influence*

Avogadro’s hypothesis was expressed in the same way as Boyle’s empirical gas law in the year 1662, Charles’s law in the year 1787 and Gay-Lussac’s law in the year 1808. The Avogadro hypothesis was published by the famous scientist Amadeo Avogadro in the year 1811.

He reconciled the atomic theory of Dalton with an “incompatible” view of other scientists like Gay-Lussac and Joseph Louis that stated a few gases were compounds of different types of fundamental molecules in integer proportions.

In the year 1814, Avogadro, and André-Marie Ampère created the same law that resulted in similar outcomes. As the scientist Ampère was more famous in France, this hypothesis got popularly referred to as the Ampère’s hypothesis. It later became popular as Avogadro-Ampère hypothesis or Ampère-Avogadro hypothesis.

*Let’s Learn The Concept In Detail*

Soon after Dalton’s atomic theory was proposed, it was found that certain experimental facts regarding the chemical combination of gases could not be properly reconciled with his theory.

It was observed by the French chemist Joseph Gay-Lussac that when two or more gases combined chemically to produce one or more gaseous substances, the volumes of the reacting gases and those of the product gases were always in simple proportions. Thus according to Gay-Lussac, two volumes of hydrogen gas combined with one volume of oxygen gas to produce two volumes of water vapour. This could not be reconciled with Dalton’s idea that one atom of hydrogen combined with one atom of oxygen to produce one molecule of water.

In order to explain Gay-Lussac’s observations, the Italian physicist L.R.A. Avogadro proposed his now-famous Avogadro hypothesis, in1811, which can be stated as follows.

Equal volumes of different gases contain equal numbers of molecules at the same temperature and pressure.

Thus if on the basis of Avogadro hypothesis it is assumed that there are n molecules in each one-litre volume of a gas at a given temperature and pressure, then according to Gay-Lussac’s observations, 2n molecules of hydrogen contained in two litres of hydrogen gas combine with n molecules of oxygen present in 1 litre of oxygen gas to produce 2n molecules of water contained in 2 litres of water vapour which is produced. Hence two molecules of hydrogen combine with one molecule of oxygen to produce two molecules of water vapour according to the following equation

2H₂ + O₂ = 2H₂O

The above equation tells us that each molecule of hydrogen and oxygen contains two atoms of these elements respectively while each molecule of water vapour contains three atoms-two of hydrogen and one of oxygen. Experimentally it is found that 16 kg of oxygen gas combine with 2.016 kg of hydrogen gas to produce 18.016 kg of water vapour. Since in each molecule of water vapour, there are two atoms of hydrogen present in combination with one atom of oxygen, the ratio of the atomic weights of oxygen and hydrogen is

Since there are two atoms of oxygen present in each molecule, the molecular weight of oxygen may be taken to be 32.00. Hence the molecular weight of hydrogen will be 2.016. Chemists define one mole of a substance as the weight of the substance equal to its molecular weight expressed in grams i.e 10^{-3} kg.

The volume of one mole of a gas is known as the molar volume or the gram-molecular volume. Careful measurements have established that at standard temperature (0°C) and pressure (760 mm of Hg), one molar volume of a gas is V=22.4136 litres. According to the Avogadro hypothesis, the number of molecules in each mole of any gas is the same. This is therefore a universal constant and is known as the Avogadro number, usually designated by the symbol N_{0}.

From the above discussion, it is obvious that there are N_{0} molecules of oxygen in 32 × 10^{-3} kg or 32g of oxygen gas. So there are 2N_{0} oxygen atoms in it. Hence in 16 x 10^{-3} kg or 16 g of oxygen, which is the atomic weight of oxygen expressed in grams, there are N_{0} atoms of oxygen. So in each gram-atomic weight of an element, the number of atoms is equal to the Avogadro number N_{0}. From this, the mass of each atom of the element, expressed in gram, can be determined by dividing the gram-atomic weight by N_{0}. This shows clearly the importance of determining the Avogadro number very accurately.

*What is Avogadro’s Number?*

Avogadro’s number is defined as the number of units in one mole of a substance. This consideration is equal for both the lightest gas (Hydrogen) and heavy gases (Bromine or Carbon Dioxide). The value of Avogadro’s Number is 6.022×10^{23}.

*How Many Numbers of Grams Are Present in One Mole?*

There are around 6.022×10^{23} atoms present in one mole or per twelve grams of carbon. The number of moles present in any substance can be easily found by using the molar mass of that substance. This mass is the number of grams present in a single mole of any substance.

Taking STP to be 101.325 kPa and 273.15 K, we can find the volume of one mole of gas: