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## Definition of Avogadro’s Hypothesis (Avogadro’s Law)

Avogadro’s hypothesis states that equal volumes of different gases contain an equal number of molecules under the same temperature and pressure conditions.

This empirical relation can be derived from the kinetic theory of gases under the assumption of a perfect (ideal) gas. The law is approximately valid for real gases at sufficiently low pressures and high temperatures. This law is also known as Avogadro’s hypothesis.

### Other Definitions

Some more definitions of this law are listed below to get more clarity about this law:

### Definition-1

The law was first proposed in 1811 by Amedeo Avogadro, a professor of higher physics at the University of Turin, for many years. But this law did not generally accept it until after 1858, when an Italian chemist, Stanislao Cannizzaro, constructed a logical chemistry system.

### Definition-2

Avogadro’s hypothesis is another classical gas law. It can be stated: At the same temperature and pressure, equal volumes of different gases contain the same number of molecules.

When the mass, in grams, of an ideal gas sample is equal to the gram molar mass (traditionally called the molecular weight) of the gas, the number of molecules in the sample is equal to Avogadro’s number, N¯^{1}. Avogadro’s number is the number of molecules in a mole.

In the modern definition, one mole is the number of atoms of C^{12} in exactly 12g of C^{12}. The number of atoms of C^{12} in exactly 12g of C^{12} is Avogadro’s number. The currently accepted value is 6.02214199×1023 molecules per mole. We can find the gram atomic mass of any other element by finding the mass of that element that combines with exactly 12g of C^{12} in a compound whose molecular formula is known.

### Definition-3

The validity of Avogadro’s hypothesis follows immediately either from the fact that the Boyle’s law constant, α(T), is the same for any gas or from the fact that the Charles’ law constants, β(P) and γ(P), are the same for any gas.

However, this entails a significant circularity; these experiments can show that α(T), β(P), and γ(P) are the same for any gas-only if we know how to find the number of moles of each gas that we use. To do so, we must know the molar mass of each gas. Avogadro’s hypothesis is crucially important in the history of chemistry: Avogadro’s hypothesis made it possible to determine relative molar masses. This made it possible to determine molecular formulas for gaseous substances and to create the atomic mass scale.

## Formula of the Avogadro’s Hypothesis

At constant pressure and temperature, Avogadro’s law can be expressed via the following formula:

V ∝ n

V/n = k

Where V is the volume of the gas, n denotes the amount of gaseous substance (often expressed in moles), and k is a constant.

When the amount of gaseous substance is increased, the corresponding increase in the volume occupied by the gas can be calculated with the help of the following formula:

V_{1}/n_{1} = V_{2}/n_{2} = k, (as per Avogadro’s law).

## Derivation of the Avogadro’s Law

Avogadro’s law can be derived from the ideal gas equation, which can be expressed as follows:

PV = nRT

Where,

‘P’ is the pressure exerted by the gas on the walls of its container

‘V’ is the volume occupied by the gas

‘n’ is the amount of gaseous substance (number of moles of gas)

‘R’ is the universal gas constant

‘T’ is the absolute temperature of the gas

Rearranging the ideal gas equation, the following equation can be obtained.

V/n = (RT)/P

Here, the value of (RT)/P is a constant (since the temperature and pressure kept constant and the product/quotient of two or more constants is always a constant).

Therefore:

V/n = k

Thus, the proportionality between the volume occupied by a gas and the number of gaseous molecules is verified.

## History of Avogadro’s Hypothesis

Avogadro’s hypothesis (as it was known originally) was formulated in the same spirit of earlier empirical gas laws like Boyle’s law (1662), Charles’s law (1787) and Gay-Lussac’s law (1808).

Amadeo Avogadro first published the hypothesis in 1811. It reconciled Dalton atomic theory with the “incompatible” idea of Joseph Louis Gay-Lussac that some gases were composite of different fundamental substances (molecules) in integer proportions.

In 1814, independently from Avogadro, André-Marie Ampère published the same law with similar conclusions. As Ampère was more well known in France, the hypothesis was usually referred to there as Ampère’s hypothesis, and later also as Avogadro–Ampère hypothesis or even Ampère–Avogadro hypothesis.

## Significance of Avogadro’s Law

- 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, diatomic molecules of elements such as H
_{2}, O_{2}, and C_{12}were not recognized until the results of experiments involving gas volumes were 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 gives 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}gives 2HCl.

## Avogadro’s Constant

- The specific number of molecules in one gram-mole of a substance, defined as the molecular weight in grams, is 6.02214076 × 1023, a quantity called Avogadro’s number, or the Avogadro constant. For example, the molecular weight of oxygen is 32.00, so that one gram-mole of oxygen has a mass of 32.00 grams and contains 6.02214076 × 1023 molecules. The volume occupied by one gram-mole of gas is about 22.4 litres (0.791 cubic foot) at standard temperature and pressure (0 °C, 1 atmosphere) and is the same for all gases, according to Avogadro’s law.
- Avogadro’s law provides a way to calculate the quantity of gas in a receptacle. Thanks to this discovery, Johann Josef Loschmidt, in 1865, was able for the first time to estimate the size of a molecule. His calculation gave rise to the concept of the Loschmidt constant, a ratio between macroscopic and atomic quantities.
- In 1910, Millikan’s oil drop experiment determined the electron’s charge; using it with the Faraday constant (derived by Michael Faraday in 1834), one can determine the number of particles in a mole of the substance. At the same time, precision experiments by Jean Baptiste Perrin led to the definition of Avogadro’s number as the number of molecules in one gram-molecule of oxygen. Perrin named the number to honour Avogadro for his discovery of the namesake law. Later standardization of the International System of Units led to the modern definition of the Avogadro constant.

## Applications of Avogadro’s Hypothesis

- Explaining Gay Lussac’s law of gaseous volumes.
- In determining the atomicity of gasses.
- In determining the molecular formula of a gas.
- Establishing the relationship between relative molecular mass and vapour density.

## Limitations of Avogadro’s Hypothesis

- Despite being perfectly applicable to ideal gases, Avogadro’s law provides only approximate relationships for real gases. The deviation of real gases from ideal behaviour increases at low temperatures and high pressures.
- It is important to note that gases molecules having relatively low molecular masses (such as helium and hydrogen) obey Avogadro’s law to a greater extent than heavier molecules.

## Avogadro’s Law in Everyday Life

You have probably experienced this example of Avogadro’s Law yourself. When you blow up a balloon, you are adding molecules of gas into it.

The result is that the volume of the balloon increases – and to do this, you decrease the number of molecules in your lungs (which decreases their volume)! A bicycle pump does the same thing to a bicycle tire.

## Examples of Avogadro’s Hypothesis (Avogadro’s Law)

- The process of respiration is an excellent example of Avogadro’s law. When humans inhale, then the molar quantity of air increase in the lungs is accompanied by an increase in the volume of the lungs. An image detailing the change in volume brought on by an increase in gaseous molecules is provided below.
- Another typical example of Avogadro’s law is the deflation of automobile tyres. When the air trapped inside the tire escapes, the number of moles of air present decreases. This decreases the volume occupied by the gas, causing the tyre to lose its shape and deflate.

## Solved Questions

Here are some solved questions based on Avagadro’s Law:

### Question-1

One mole of helium gas fills up an empty balloon to a volume of 1.5 liters. What would be the volume of the ballon if we add extra 2.5 moles of helium gas?

**Solution:** (Assume that the temperature and the pressure are kept constant)

Given,

The initial amount of helium (n_{1}) = 1 mol

The initial volume of the balloon (V_{1}) = 1.5 L

The final amount of helium (n_{2}) = 1 mol + 2.5 mol = 3.5 mol

As per Avogadro’s law,

V1/n1 = V2/n2

Therefore,

the final volume of the balloon (V_{2}) = (V_{1}n_{2})/n_{1} = (1.5L*3.5mol)/1mol = 5.25 L

The balloon would occupy a volume of 5.25 liters when it contains 3.5 moles of helium gas.

### Question-2

A tyre containing 10 moles of air and occupying a volume of 40L loses half its volume due to a puncture. Considering that the pressure and temperature remain constant, what would be the amount of air in the deflated tyre?

**Solution:** Given,

The initial amount of air (n_{1}) = 10 mol

The initial volume of the tyre (V_{1}) = 40 L

The final volume of the tyre (V_{2}) = 20 L

According to Avogadro’s law,

The final amount of air in the tyre (n_{2}) = (V_{2} n_{1})/V_{1} = 5 moles.

The deflated tyre would contain 5 moles of air.

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