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Avogadro’s Law

Avogadro’s law formula, applications and uses [1], problems and solutions.

Avogadro’s law states that equal volumes of different gases contain an equal number of molecules under the same pressure and temperature conditions. This law is valid for ideal gases at low pressures and high temperatures [1-4] .

Italian physicist Amedeo Avogadro was the first to state the hypothetical law in 1811.

According to Avogadro’s hypothesis, the volume (V) of a gas is directly proportional to the number of moles (n) [1-4] .

n : Number of moles

The above proportionality can be written as follows:

k : proportionality constant

This equation can be represented as a graph.

Consider a gas with n 1 moles with a volume V 1 . Suppose the number of moles increases to n 2 such that the volume increases to V 2 , then

V 1 = kn 1 and V 2 = kn 2

Dividing one equation by the other and rearranging,

V 1 /n 1 = V 2 /n 2

The above equation can be used to compare two gases at the same temperature and pressure conditions. Also, the number of molecules in a given volume of an ideal gas is independent of their size or the molar mass.

In order to derive Avogadro’s gas law, let us look at the ideal gas equation.

Or, V/n = RT/P

P : Pressure

R : Universal gas constant

T : Temperature

At constant pressure and temperature, the right-hand side is constant. Let, k = RT/P. Then,

Thus, the volume is proportional to the number of moles of the gas.

Avogadro’s Number

Let us rewrite the ideal gas law as follows:

PV = (N/N A )RT

The ratio N/N A gives the number of moles or n = N/N A . Here, N is the number of molecules in the gas, and N A is known as Avogadro’s number. It is the number of molecules present in one mole of a substance. Its value is 6.023 x 10 23 .

Molar Volume

Avogadro’s number can be used to calculate the molar volume. For one mole of a gas, n = 1 and V = k. Therefore, one has to know the value of k at standard temperature and pressure (STP). At STP, the pressure (P) is 101.325 kPa, and the temperature (T) is 273.15 K. The universal gas constant R has a value of 8.314 L·kPa·M -1· K -1 . Putting in all the values, we get

Or, k = 8.314 L·kPa·M -1· K -1 x 273.15 K /101.325 kPa

Or, k = 22.4 LM -1

Below are some examples of Avogadro’s law in everyday life [5] .

• Respiration : When we respire, we breathe in oxygen. The more we breathe, the more our lungs will expand.
• Deflation : When the air is released from an inflated tire, the number of moles decreases. The shape of the tire also changes since its volume decreases.
• Inflation : When a water tube is inflated by pumping air inside, the number of air molecules increases. Thus, the tube is inflated, and its volume increases.
• Explains Gay-Lussac’s law
• Determines the atomicity of the gases
• Determines the molecular formula of a gaseous compound
• Gives the relationship between gram molecular mass and gram molecular volume of gas
• Determines the relationship between molecular mass and vapor density of a gas

Problem 1 : Two moles of helium gas fill up an empty balloon to a volume of 2.5 L. What would be the volume of the balloon if an additional 1.5 moles of helium gas is added at constant temperature and pressure.

n 1 = 2 mol

V 1 = 2.5 L

n 2 = 2 mol + 1.5 mol = 3.5 mol

From Avogadro’s law,

Therefore, the final volume of the balloon is

 V 2 = V 1 ·n 2 /n 1 = (2.5 L x 3.5 mol)/2 mol = 4.375 L

Problem 2 : 40 g of nitrogen gas is kept in a 2.5 L container. The gas exerts a pressure of 2 atm on the container. If pressure is kept constant, what is the final amount of gas in grams present in the container if gas is added until the volume has increased to 4.0 L?

Mass of nitrogen = 40 g => n 1 = Mass/Molar mass = 40 g/28 gM -1 = 1.43 M

Or, n 2 = n 1 x V 2 /V 1

Or, n 2 = 1.43 M x 4 L/2.5 L

Or, n 2 = 2.29 M

Therefore, amount of nitrogen

= 2.29 M x 28 gM -1 = 64 g

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Avogadro’s Hypothesis: Law: Example: Application

• August 7, 2022
• Gaseous State

Table of Contents

Avogadro’s Hypothesis , introduced by Amaedo Avogardro in 1811, explains that in every gas the number of molecules is proportional to the volume. He replaced the term ‘atoms’ in the Berzelius hypothesis with the term ‘molecules’ and postulated a law, popularly known as Avogadro’s hypothesis.

Avogadro’s hypothesis states, “ Equal volumes of all gases and vapors contain an equal number of molecules under similar conditions of temperature and pressure .” According to this hypothesis, under the same temperature and pressure, the volume of a gas depends upon the number of molecules or moles but not upon their size or mass. Under the assumption of an ideal (perfect) gas , this empirical relationship can be derived from the kinetic theory of gases .

This law is valid for real gases at low pressure and high temperature.

Example of Avogadro’s law

If we consider 2 liters of nitrogen, carbon dioxide, and hydrogen each under STP, they will contain an equal number of molecules however their mass is different. This hypothesis is experimentally verified and is called Avogadro’s law . Avogadro’s law can explain gaseous reactions without violating Dalton’s atomic theory .

Let 1 volume of gas contain ‘n’ molecule then apply Avogadro’s law

The above experimental result shows that 1 molecule of hydrogen chloride is produced from 1/2 molecule of hydrogen and half molecule of chlorine. The fraction of molecules is possible because a molecule contains more than one atom . Hence Avogadro’s hypothesis is in accordance with dalton’s atomic theory and Gay-Lussac’s law of gaseous volume.

Application of Avogadro’s law

Avogadro’s law has numerous applications, some of which are as follows:

1. Determination of atomicity of elementary gases

Atomicity is the number of atoms present in one molecule of an element or compound. elementary gases are composed of atoms of the same element. For example, H 2 , Cl 2 , O 2 , N 2 , etc. Other types of gases are compound gases like CO 2 , NH 3 , SO 2 , N 2 O 5 , etc. Avogadro’s law is applied to determine the atomicity of elementary gases. Elementary gases are diatomic, i.e., one molecule contains two atoms. For Further illustration, we can take the example of oxygen.

1.1.Oxygen is Diatomic gas

Oxygen reacts with hydrogen to give water vapor. Experimentally, it has been found that hydrogen and oxygen react in the ratio of 2:1 by volume to give 2 volumes of water vapor under similar conditions of temperature and pressure. This experimental result is formulated as follows:

Let one volume of gas contain ‘n’ molecules, then by applying Avogadro’s hypothesis,

One molecule of water vapor contains one molecule of hydrogen and 1/2 molecule of oxygen. From Avogadro’s law , it has been proved that one molecule of hydrogen contains 2 atoms. The valency of hydrogen is one. Hence, two atoms of hydrogen can satisfy the valency of only one atom of oxygen because the valency of oxygen is two. From the above discussion, we come to know that one molecule of water vapor contains only one atom of oxygen. This one atom is derived from its half molecule. Hence, we can write,

1/2 molecule of oxygen contains 1 atom or 1 atom of oxygen contains 2 atoms.

Hence, by applying Avogadro’s hypothesis we are able to determine the atomicity of oxygen i.e, 2.

2. Deduction of the relationship between molecular mass and vapor density

Molecular mass or molecular weight is the mass of one molecule of a substance as compared to the mass of one atom of hydrogen.

From Avogadro’s law, we know that hydrogen is diatomic i.e. a molecule containing two atoms. Hence, we can write

Mol. wt. = 2 × V.D.

Hence, the molecular weight of a gaseous substance is twice its vapor density .

3. Derivation of the relationship between gram molecular mass and volume of gas

The molecular weight expressed in grams is called gram molecular mass. Experimentally, it has been found that the gram molecular mass of any gas or vapor at STP occupies 22.4 liters. This experimental observation is theoretically explained by applying Avogadro’s law .

4. Derivation of the relationship between gram molecular weight and number of molecules

From Avogadro’s law, we know that the gram molecular weight of any gas or vapor occupies 22.4 liters at STP. This quantity is also called molar volume i.e, the volume occupied by one molecule. According to Avogadro’s law, we know that the equal volume of all gases and vapors contains an equal number of molecules under similar conditions of temperature and pressure.

Since one mole of any gas or vapor at STP occupies 22.4 liters. Due to equal volume, it must contain the same number of molecules. Experimentally it has been found that 22.4 liters of gas at STP contain 6.023 × 10 23 molecules.

5. Determination of molecular formula from volume composition of a gas

Avogadro’s law is also applied to determine the molecular formula of a gaseous compound from its volume composition and molecular mass. It can be explained by considering an oxide of nitrogen having a molecular mass of 60 and one volume of which is composed of half of the volume of nitrogen and half of the volume of oxygen i.e. volume ratio of nitrogen: oxygen: nitrogen oxides is 1:1:2 under the similar condition of temperature and pressure.

What is Avogadro’s hypothesis?

Avogadro’s hypothesis states, “ Equal volumes of all gases and vapors contain an equal number of molecules under similar conditions of temperature and pressure .”

What evidence supports Avogadro’s hypothesis?

The fraction of molecules is possible because a molecule contains more than one atom.

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• Avogadros Hypothesis

A Tale of Temperature and Pressure Conditions!

Avogadro's law states that if the condition of temperature and pressure are the same, then the equal volume of different gases also will have the same number of molecules in it. You can derive this from the kinetic theory of gas under the assumption of taking a perfect or an ideal gas. This law is more appropriate and applicable for gases at relatively low pressure and high temperature.

Now, if we talk about Avogadro's Number then it is the sacrifice number of molecules in the Gram mole of a substance, it can be defined as molecular weight in grams it is 6.02214706 ×10 23 . 

This law was designed by professor Avogadro of higher physics at the University of Turin. Although he proposed this law in 1811, this law was not accepted upto the year 1858 when an Italian chemist namely Stanisalo Cannizaro constructed a whole logical system of chemistry based on this law. The more information about this law, you will get in this article below. Let's have a look at it.

Avogadro's Hypothesis - Avogadro’s Constant

The law of Avogadro provides a means to measure the amount of gas present in a receptacle. This law is sometimes even called Avogadro's principle or Avogadro's hypothesis. It is a kind of an experimental gas law that relates the gas volume to the quantity of gaseous substance present.

It is a precise case of an ideal gas law. The law of Avogadro states that when all gases of equal volumes are mixed at the same pressure and temperature, then it has the same amounts of molecules.

For any specific weight of a gas, the amount and volume of the gas substance is proportional to each other provided that both the pressure and temperature remains constant.

So, the entire law of Amedeo Avogadro concluded that different types of gases bearing the same volume, when combined at the same pressure and temperature, possess the same amount of in it. 

For example, when the two ideal gases, hydrogen and nitrogen are mixed in equal quantities, they contain the same amounts of molecules in them. This happens only when they are kept at the same pressure and temperature. This shows the ideal behaviour of gas. 

Now Let Us See this Law in Mathematical Terms:

The entire law can be written as V \[\infty\] n

\[\frac{V}{n} = K\] (a constant)

‘V’ is the symbol for gas volume;

‘n’ is the gaseous substance that is measured in the form of moles;

‘k’ is a constant defined for a particular pressure and temperature.

When compared the same gaseous substance under different conditions, this law can be written as:

\[\frac{V_{1}}{n_{1}} = \frac{V_{2}}{n_{2}}\]

So, the above equation shows that the quantity of moles in the gas increases in the same proportion as the increase in the gas volume. If the amount of gas moles decreases, then its volume also reduces. So, the total number of atoms or molecules present in any particular volume of gas is completely independent of the molar mass or size of the gas.

How the Avagadro’s Hypothesis Got Formed?

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 compound 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.

What Problems Did Avogadro Face Before His Hypothesis Got Accepted by People?

Chemists found the Avogadro hypothesis quite tough to accept as there were some things that were not easy to accept. One of the things was that if the gaseous particles were combining, then a volume of gas X when reacted with a volume of gas Y should result in one product volume.

Later after several researches, it was found that a volume of gas X when reacted with a volume of gas Y resulted in the formation of two product volumes. Chemists such as Gay-Lussac were not able to interpret this type of gaseous behavior. Avogadro also reasoned that such an outcome can only be defined when Gay-Lussac’s particles were actually made up of tiny particles. He then introduced the application of the term “molecule” in order to refer to the combinations of small sized particles. 

The word “molecule” was derived from the term “mole”. The term “mole” implied “lumps of matter” and the term “cula” implied “little”. So, a molecule was seen as a tiny cluster of matter. The hypothesis given by Avogadro was totally consistent with the assessment that the pressure was caused due to the molecular collisions with the different sides of the container. 

On keeping the pressure the same, it was pretty reasonable that the amounts of molecules contained in the container also remained the same. Along with it, when hydrogen chloride and ammonia were mixed in equal ratio of one to one, he arrived at the conclusion that a number of different gases mixed in this manner can only happen if the individual gas molecules were “diatomic”.

In What Way Did Avogadro's Hypothesis Help Chemists?

This hypothesis stated that two different gas samples of equal volume, when present at the same pressure and temperature, contain the equal number of molecules. Avogadro's hypothesis permitted chemists to forecast the performance of an ideal gas.

Amedeo Avogadro created the hypothesis in the year 1811, in which he asserted that the gas volume is independent of the mass or size of the gas molecules. He further stated that equal gas volumes present at the same pressure and temperature contain the equal number of molecules irrespective of their physical properties and chemical nature. This number is referred to as Avogadro's number.

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 .

Why is Avogadro Hypothesis Important?

Avogadro's law established how the gas amount (n) is related to its volume (v). It was found to be a direct relationship, which implied that the gas volume is completely proportional to the amount of gas moles present in it. This law was very significant as it helped in saving a lot of money and time in the long run.

What is the Concept of Mole and How it Was Calculated?

Mole is described as the quantity of any substance that is made up of as many molecules, atoms, electrons, ions, or any different forms of objects in 12 grams of carbon. This quantity is referred to as Avogadro's number.

What is a Molar Mass?

The mass of one mole of a substance is referred to as its molar mass. This mass is used for the conversion of specific grams of any substance to its moles. It has applications in the field of chemistry too. If one knows the mass of any substance, then the quantity of moles contained in it can be easily calculated.

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.

Avogadro wasn’t able to base the hypotheses about the diatomic characteristics of molecules. He illustrated several experiments but the scientific community could not accept the validity of the hypothesis till several years post his death. Using the hypothesis, it assisted the chemists to deduce the chemical formulas of gaseous substances on the basis of combining gas volumes. The application of Avogadro’s hypothesis was instrumental in defining the molar element’s mass. Students can go through concepts, Definitions, and questions carefully and understand the concepts used to solve these questions. This will help the students immensely in their examinations. 

FAQs on Avogadros Hypothesis

1. What Does Avogadro's law state?

Avogadro's law, in simple language, states that an equal volume of different gases also contains the same number of molecules provided that the condition of temperature and pressure also remains the same. In most of these conditions, Avogadro's law is applicable when the temperature is high and pressure is low. This law initially was not accepted but when an Italian scientist logically explained chemistry on its basis, then this law was proved correct. Avogadro law is now a fundamental law accepted by scientists all around the world. 

2. What is the importance of Avogadro’s law?

Avogadro’s law states that an equal volume of gases at the same temperature and pressure conditions will have equal molecules of gas. It is important because it serves as a link between the amount of gas and the volume of gas. This is a direct relationship that states that the volume of gas is directly proportional to the number of moles present in the gas sample at a temperature and pressure. This law is quite significant because it allows you to save time and money, and you are not required to spend a lot of energy. 

3. State Avagadro’s law and give examples of Avogadro's law?

Avogadro's law is simple. It states that if different gases have the same volume and are under the same temperature and pressure conditions, then the number of molecules in the gases will be the same. Let us understand this topic by using simple examples. We all know about respiration. The respiration process is the best example of Avogadro's law. When a person inhales the air, the increase in the molar quantity of the lungs is accompanied by an increase in the volume capacity. During letting out of the air, the decrease in the molar quantity of air in the lungs will lead to decrease in the volume of air in the lungs and thus, lead to a decrease in the lungs capacity.

4. Why is Avogadro’s law only applicable for gases?

Avogadro's law or Avogadro's hypothesis is mostly applicable only for gases. This is because there is much space present between two molecules of a gas. The space between them is so much that the size of the molecule has no bearing on the volume of the material of the gas. This is the main reason why the volume of gas is governed by the pressure applied to the volume of gas, and under the same pressure, the gases have the same volume. This is what Avogadro's law states that under the same pressure, there will be the same number of molecules in two different gases.

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Avogadro’s Hypothesis: Avogadro’s Law, Examples, Formula, Applications

You will find the answer to all these interesting questions in this article about Avogadro’s Hypothesis . The atomicity of water \(\left( {{{\rm{H}}_2}{\rm{O}}} \right)\) is \(3\). A molecule of water is made up of \(2\) hydrogen atoms and \(1\) oxygen atom. Now, how many hydrogen and oxygen molecules make up a molecule of water? At constant temperature and pressure, is the number of molecules in different gases like hydrogen, oxygen, steam, ammonia, etc. the same or different?

The law is named after Amedeo Avogadro, who suggested in 1812 that two identical samples of an ideal gas with the same volume, temperature, and pressure have the same number of molecules. When equal amounts of gaseous hydrogen and nitrogen are at the same temperature and pressure, they contain the same number of atoms and exhibit perfect gas behaviour.

In this article, you will explore Avogadro’s law, examples of it, Avogadro’s constant its applications in detail and more. Continue reading for more information.

What is Avogadro’s Hypothesis?

The Italian chemist Amedeo Avogadro, established a relationship between the volume of a gas and the corresponding number of molecules under a given set of conditions of temperature and pressure. This hypothesis is called Avogadro’s hypothesis. It states that, under similar conditions of temperature and pressure, an equal volume of all gases contain an equal number of molecules.

\({\rm{V}} \propto {\rm{N}}\)

Where V is the volume of the gas and N is the number of molecules.

For example, if an equal volume of four gases hydrogen \(\left( {{{\rm{H}}_2}} \right)\), oxygen \(\left( {{{\rm{O}}_2}} \right)\), chlorine \(\left( {{\rm{C}}{{\rm{l}}_2}} \right)\) and ammonia \(\left( {{\rm{N}}{{\rm{H}}_3}} \right)\) is enclosed in the different flasks of the same capacity under similar conditions of temperature and pressure, then all flasks have the same number of molecules. However, these molecules may be different in size and mass.

Avogadro’s Hypothesis and Dalton’s Atomic Theory

Avogadro’s Hypothesis is a modification of the Berzelius hypothesis. According to the Berzelius hypothesis, an equal volume of all gases under similar conditions of temperature and pressure contains an equal number of atoms.

But this hypothesis is not applicable to the chemical reactions involving gases. It was found that even fractions of atoms were involved in some chemical reactions. But in the Dalton atomic theory half of an atom of an element cannot exist. This conflict was solved by Avogadro by making a clear distinction between atom and molecule.

According to Avogadro, an atom is the smallest particle of an element that may or may not have an independent existence. In contrast, a molecule is the smallest particle of a substance (element or compound) that can exist independently.

L earn Ideal Gas Equation

Example for Avogadro’s Hypothesis: Formation of HCl Gas

One volume of hydrogen and one volume of chlorine combine to give two volumes of hydrogen chloride gas and NTP (Normal Temperature Pressure) conditions.

\({\rm{Hydrogen}} + {\rm{Chlorine}} \to {\rm{Hydrogen}}\,{\rm{chloridegas}}\)

\(1\,{\rm{Volume}}\,1\,{\rm{Volume}}\,2\,{\rm{Volumes}} = {\rm{kn}}\)

Let \(1\) volume of each gas contain n molecules.

By applying Avogadro’s hypothesis

\({\rm{Hydrogen}} + {\rm{Chlorine}} \to {\rm{Hydrogen}}\,{\rm{chloridegas}}\) \({\rm{n}}\,{\mkern 1mu} {\rm{molecules}}\,\,{\rm{n}}\,{\mkern 1mu} {\rm{molecules}}{\mkern 1mu} \,\,2{\rm{n}}{\mkern 1mu} \,{\rm{molecules}}\) \(1\,{\mkern 1mu} {\rm{molecules}}\,{\mkern 1mu} 1\,{\mkern 1mu} {\rm{molecules}}{\mkern 1mu} \,2{\mkern 1mu} \,{\rm{molecules}}\) \(\frac{1}{2}{\mkern 1mu} \,{\rm{molecules}}{\mkern 1mu} \,\frac{1}{2}{\mkern 1mu} \,{\rm{molecules}}{\mkern 1mu} \,1{\mkern 1mu} \,{\rm{molecules}}\)

This means that \(1\) molecule of hydrogen chloride contains ½ molecule of hydrogen and \(1/2\) molecule of chlorine. Now, \(1/2\) molecule of hydrogen can exist because one molecule of hydrogen contains two atoms of hydrogen, and \(1/2\) molecules of hydrogen mean one atom of hydrogen. Similarly, \(1/2\) molecule of chlorine contains an atom of chlorine because chlorine is also a diatomic molecule. Thus, one molecule of hydrogen chloride is formed from \(1\) atom of hydrogen and \(1\) atom of chlorine. This agrees with Dalton theory.

What is the Value of Avogadro Constant?

The number of molecules in one mole of gas has been determined to be \(6.022 \times {10^{23}}\). This value is known as Avogadro Constant.

According to Avagadro, all gases containing an equal amount of substances occupy the same volume at the same temperature and pressure.

\({\rm{V}} \propto {\rm{n}}\)

\({\rm{V = kn}}\)

Where \({\rm{k}}\) is the proportionality constant.

One mole, each gas at standard temperature and pressure, will have the same volume. This is known as molar volume \(\left( {{{\rm{V}}_{\rm{m}}}} \right)\). \(1\) mole of any gas at the \(273.15\;{\rm{K}}\) and \(1\) bar pressure occupies \({22.7110^{ – 3}}\;{{\rm{m}}^3}\) or \(22.71\;{\rm{L}}\).

The number of moles can be calculated by the equation,

We know that, \({\rm{n}} = \frac{{{\rm{Mass}}\,{\rm{of}}\,{\rm{gas}}}}{{{\rm{Molar}}\,{\rm{mass}}}} = \frac{{\rm{m}}}{{\rm{M}}}\)

\({\rm{V}} = {\rm{k}}\frac{{\rm{m}}}{{\rm{M}}}\)

\({\rm{M}} = {\rm{k}}\frac{{\rm{m}}}{{\rm{V}}}\)

We know that density, \({\rm{d}} = \frac{{\rm{m}}}{{\rm{v}}}\)

 On rearranging,

\({\rm{M = kd}}\)

Hence, the density of a gas is directly proportional to its molar mass.

Applications of Avogadro’s Law

1. Deduction of atomicity of Elementary Gases: Atomicity of an elementary substance is defined as the number of atoms of the element present in one molecule of a  substance. Example: Atomicity of oxygen \(\left( {{{\rm{O}}_{\rm{2}}}} \right)\) is \(2\) while that of ozone \(\left( {{{\rm{O}}_{\rm{3}}}} \right)\) is \(3\). Avogadro’s law helps in determining the atomicity of elementary gases such as hydrogen, oxygen, chlorine, etc. Example: Calculation of atomicity of oxygen. Consider the reaction between hydrogen and oxygen to form water vapour. Two volumes of hydrogen combined with \(1\) volume of oxygen to form two volumes of water vapour. \({\rm{Hydrogen}} + {\rm{Oxygen}} \to {\rm{Water}}\,{\rm{Vapour}}\) \(2\,{\rm{Volumes}}\,1\,{\rm{Volume}}\,2\,{\rm{Volumes}}\) Applying Avogadro’s hypothesis \({\rm{Hydrogen}} + {\rm{Oxygen}} \to {\rm{Water Vapour }}\) \(2{\rm{n}}\,{\rm{molecules}}\,{\rm{n}}\,{\rm{molecules}}\,2{\rm{n}}\,{\rm{molecules}}\) \(1\,{\rm{molecules}}\,\frac{1}{2}\,{\rm{molecules}}\,1\,{\rm{molecules}}\) Thus, \(1\) molecule of water contains \(\frac{1}{2}\) molecule of oxygen. But \(1\) molecule of water contains \(1\) atom of oxygen. Hence, \(\frac{1}{2}\) molecules of oxygen \( = 1\) atom of oxygen Or \(1\) molecule of oxygen \(= 2\) atoms of oxygen, i.e., atomicity of oxygen \(= 2\).

2. Determination of the relationship between vapour density and molar mass of a gas: The vapour density of a gas is the ratio between the mass of a certain volume of the gas to the mass of the same volume of hydrogen gas under the similar conditions of temperature and pressure. \({\rm{Vapour}}\,{\rm{density}}\,({\rm{V}}.{\rm{D}}.)\,{\rm{of}}\,{\rm{gas}} = \frac{{{\rm{Mass}}\,{\rm{of}}\,{\rm{certain}}\,{\rm{volume}}\,{\rm{of}}\,{\rm{gas}}}}{{{\rm{Mass}}\,{\rm{of}}\,{\rm{same}}\,{\rm{volume}}\,{\rm{of}}\,{\rm{hydrogen}}}}\) According to Avogadro’s hypothesis, equal volume of all gases under similar conditions of temperature and pressure contains equal number of molecules. Let the given volume of the gas and hydrogen contain n molecule at STP conditions. \({\rm{Vapour}}\,{\rm{density}} = \frac{{{\rm{Mass}}\,{\rm{of}}\,{\rm{n}}\,{\rm{molecules}}\,{\rm{of}}\,{\rm{gas}}}}{{{\rm{Mass}}\,{\rm{of}}\,{\rm{n}}\,{\rm{molecules}}\,{\rm{of}}\,{\rm{hydrogen}}}}\) \({\rm{Vapour}}\,{\rm{density}} = \frac{{{\rm{Mass}}\,{\rm{of}}\,{\rm{1}}\,{\rm{molecules}}\,{\rm{of}}\,{\rm{gas}}}}{{{\rm{Mass}}\,{\rm{of}}\,{\rm{1}}\,{\rm{molecules}}\,{\rm{of}}\,{\rm{hydrogen}}}}\) The ratio of the mass of one molecule of gas to the mass of an atom of hydrogen is called molar mass. Therefore, \({\rm{Vapour}}\,{\rm{density}} = \frac{{{\rm{Molar}}\,{\rm{Mass}}}}{{\rm{2}}}\) \({\rm{Molar}}\,{\rm{Mass}} = 2 \times {\rm{Vapour}}\,{\rm{density}}\) Vapour density is also called relative density of the gas.

3. Determination of relationship between mass and volume of the gas:

\({\rm{Molar}}\,{\rm{Mass}} = 2 \times {\rm{Vapour}}\,{\rm{density}}\)

\({\rm{Molar}}\,{\rm{Mass}} = 2 \times \frac{{{\rm{Mass}}\,{\rm{of}}\,{\rm{certain}}\,{\rm{volume}}\,{\rm{of}}\,{\rm{gas}}\,{\rm{at}}\,{\rm{STP}}}}{{{\rm{Mass}}\,{\rm{of}}\,{\rm{same}}\,{\rm{volume}}\,{\rm{of}}\,{\rm{hydroen}}\,{\rm{at}}\,{\rm{STP}}}}\)

\({\rm{Molar}}\,{\rm{Mass}} = 2 \times \frac{{{\rm{Mass}}\,{\rm{of}}\,1{\rm{L}}\,{\rm{of}}\,{\rm{gas}}\,{\rm{at}}\,{\rm{STP}}}}{{{\rm{Mass}}\,{\rm{of}}\,1{\rm{L}}\,{\rm{of}}\,{\rm{hydroen}}\,{\rm{at}}\,{\rm{STP}}}}\)

But, the mass of \({1{\rm{L }}}\) of hydrogen gas is \({\rm{0}}{\rm{.089 g}}\)

Therefore, \({\rm{Molar}}\,{\rm{Mass}} = 2 \times \frac{{{\rm{Mass}}\,{\rm{of}}\,1{\rm{L}}\,{\rm{of}}\,{\rm{gas}}\,{\rm{at}}\,{\rm{STP}}}}{{0.089}}\)

\({\rm{Molar}}\,{\rm{Mass}} = \frac{2}{{0.089}} \times {\rm{Mass}}\,{\rm{of}}\,1{\rm{L}}\,{\rm{of}}\,{\rm{gas}}\,{\rm{at}}\,{\rm{STP}}\)

\({\rm{Molar}}\,{\rm{Mass}} = 22.4 \times {\rm{Mass}}\,{\rm{of}}\,1{\rm{L}}\,{\rm{of}}\,{\rm{gas}}\,{\rm{at}}\,{\rm{STP}}\)

\({\rm{Molar}}\,{\rm{Mass}} = {\rm{Mass}}\,{\rm{of}}\,22.4{\rm{L}}\,{\rm{of}}\,{\rm{gas}}\,{\rm{at}}\,{\rm{STP}}\)

Thus, \(22.4{\rm{L}}\) of any gas at \({\rm{STP}}\) weighs equal to the molar mass of gas expressed in grams. This is called gram molecular volume.

You will be able to recollect Avogadro’s hypothesis, Avogadro’s law, and a comparison of Dalton’s theory after reading this article. Avogadro’s constant and applications of Avogadro’s hypothesis in identifying atomicity of elementary gases, determining a relationship between vapour density and molar mass of gas, and determining a relationship between mass and volume of the gas are all things you’re familiar with.

We have provided some frequently asked questions on Avogadro’s Hypothesis here:

Q.1. What is Avogadro’s equation? Ans: Avogadro’s equation is \({\rm{V}} = {\rm{kn}}\)

Where \({\rm{V}}\) is the volume of gas, \({\rm{n}}\) is the number of moles of gas and \({\rm{k}}\) is the proportionality constant.

Q.2. What are the applications of Avogadro’s Hypothesis? Ans: The applications of Avogadro’s Hypothesis are

• Avogadro’s law helps in determining the atomicity of elementary gases such as hydrogen, oxygen, chlorine, etc.
• It helps in the determination of the relationship between vapour density and molar mass of a gas. \({\rm{Vapour}}\,{\rm{density}} = \frac{{{\rm{Molar}}\,{\rm{mass}}}}{2}\)
• It helps in the determination of the relationship between mass and volume of the gas. \({\rm{Molar}}\,{\rm{mass}} = {\rm{Mass}}\,{\rm{of}}\,22.4{\rm{L}}\,{\rm{of}}\,{\rm{gas}}\,{\rm{STP}}\)

Thus, \({\rm{22}}{\rm{.4L}}\) of any gas at \({\rm{STP}}\) weighs equal to the molar mass of gas expressed in grams. This is called gram molecular volume.

Q.3. Why is Avogadro’s law important? Ans: Avogadro’s law is important for the calculation of the amount of gas present in a particular volume.

According to Avogadro’s law

Q.4. What is Avogadro’s hypothesis in chemistry? Ans: The Italian chemist Amedeo Avogadro established a relationship between the volume of a gas and the corresponding number of molecules under a given set of conditions of temperature and pressure. This hypothesis is called Avogadro’s hypothesis. It states that, under similar conditions of temperature and pressure, an equal volume of all gases contain an equal number of molecules.

Where \({\rm{V}}\) is the volume of the gas and \({\rm{N}}\) is the number of molecules.

Q.5. What does Avogadro’s law state? Ans: Avogadro’s law states that under similar conditions of temperature and pressure, the equal volume of all gases contains an equal number of molecules.

Q.6. What is Avogadro’s law example? Ans: An example for Avogadro’s Hypothesis is the formation of \({\rm{HCl}}\) gas.   One volume of hydrogen and one volume of chlorine combine to give two volumes of hydrogen chloride gas at \({\rm{NTP}}\) (Normal Temperature Pressure) condition.

\({\rm{Hydrogen}} + {\rm{Chlorine}} \to {\rm{Hydrogen}}\,{\rm{chloride}}\,{\rm{gas}}\)

\(1\,{\rm{Volmue}}\,1\,{\rm{Volume}}\,2\,{\rm{Volumes}}\)

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Why is Avogadro's hypothesis true?

Why is the number of molecules in a standard volume of gas at a standard temperature and pressure a constant, regardless of the molecule's composition or weight? Let's say I have a closed box full of a heavy gas, one meter on a side. It has a certain number of molecules inside. I want to be able to add a lighter gas to the box without changing its internal pressure (or temperature), so I connect a cylinder to the side of the box, which holds a frictionless piston for expansion (the piston has a constant force applied to it, to maintain a constant pressure inside the box and allow the volume of the gas to grow as new gas is introduced into the box). Now I add Helium to the box. The piston moves back to maintain constant pressure, but why does the number of molecules in the box proper stay constant? My mental image of this is that it would be like adding water to a bucket of marbles, and that, evidently, is wrong, but why is it wrong?

• statistical-mechanics
• physical-chemistry

• $\begingroup$ I fail to follow your construction after the third line. A figure would be extremely helpful for this question. $\endgroup$ –  user346 Commented Jan 14, 2011 at 4:32
• $\begingroup$ The number of molecules in the box actually does change. It's only in the zero-pressure limits that the experiment works the way you've described it. $\endgroup$ –  Mark Eichenlaub Commented Jan 14, 2011 at 4:33
• $\begingroup$ To me, you seem to not want to believe that the different masses of different elements, like helium oder helenium :-) do not matter. Just in case: me too, I find it incredible. $\endgroup$ –  Peter Bernhard Commented Oct 31, 2022 at 15:54

3 Answers 3

Dear Wade, your good question is easily answered if you consider "pressure" to be a derived quantity, and let us derive it.

An average molecule (or atom) of an ideal gas - and your proposition only holds for an ideal gas - has kinetic energy equal to $$mv^2/2=3kT/2$$ It's because every degree of freedom carries $kT/2$ and there are three degrees of fredom in translations. Note that lighter molecules will move faster than the heavier ones.

How do we calculate the pressure? Well, put the molecule in a cubic box of volume $a^3$. It will hit the walls - the total surface of the cube is $6a^2$. We need to compute the average (over molecules) transfered momentum per unit time - this is called the force - and divide it by the area to compute the pressure of one molecule.

The force on the wall may be in $x,y,z$ directions. Let's consider the $x$ direction. The velocity of a particular molecule in the $x$-direction is $v_x$, so it takes $a/v_x$ of time to get from one side to the other side of the box in the $x$ direction. Once it gets to the other side, it bounces off the wall and changes the sign of the $x$-component of the velocity (and momentum). At this moment, the momentum $p_x$ clearly changes the sign - i.e. changes by $2p_x$.

So the change of momentum $p_x$ per unit time is $$F_x = 2p_x / (a/v_x) = 2p_x/(am/p_x) = 2p_x^2/am$$ That's equal to $4/a$ times the kinetic energy $K_x$ in the $x$-direction. The pressure is $$p = (F_x+F_y+F_z) / 6a^2 = 3\times 4/a\times K_x / 6a^2 = 4K/a / 6a^2 = 2K/3a^3 $$ But as I have said, the average kinetic (motion) energy per molecule is $K=3kT/2$ where $T$ is the absolute temperature, so the pressure is $$p = 2k/3a^3 = kT/a^3= kT/V$$ Note that we have just derived $pV=kT$ for one molecule or $pV=NkT$ for $N$ molecules - which is what we wanted. The mass of the molecule canceled: if the molecule is heavier, the average velocity at a given temperature is slower. But that doesn't matter - because the molecule has a greater momentum (because of the higher mass) which is compensated by the longer time it needs to get from one side to the other to transfer this momentum.

So for a fixed temperature, the pressure is independent of the molecule type.

• $\begingroup$ A very good answer. I need to think it through, but I believe it. Thanks. $\endgroup$ –  user1267 Commented Feb 7, 2011 at 22:58
• $\begingroup$ Is avagadro's law true for even real vaan der Waals gasses. I guess not. Am I right $\endgroup$ –  Shashaank Commented Mar 27, 2017 at 19:06
• $\begingroup$ @Shashaank - yes, it's obviously precisely true even for non-ideal gases. This law just converts the number of atoms or molecules to moles. If you correctly define the mole as the Avogadro constant of molecules or atoms, the law is tautologically true. ... But it's not empty because at low pressures etc., every gas approaches the ideal one so there are independent ways to measure the amount of the gas in moles. So again, Avogadro's $N=n N_A$ is always true, $pV=nkT$ needs ideal gases. $\endgroup$ –  Luboš Motl Commented Apr 2, 2017 at 4:30
• $\begingroup$ But the answer just below yours says it's true only for ideal gasses. Even this one says so physics.stackexchange.com/a/129149/113699 Are they wrong ? But all real gasses at STP don't acquire 22.4 litres. I guess I am missing the basic . But then the other answers say it's true for only ideal gas.... $\endgroup$ –  Shashaank Commented Apr 2, 2017 at 6:09
• $\begingroup$ Actually there are certain such questions which are confusing. Eg : Griffith says the Current Is a Vector but answers here say it's not a vector. $\endgroup$ –  Shashaank Commented Apr 2, 2017 at 6:20

Avogadro's hypothesis holds only for ideal gases. For example, if we take a van der Waals gas with equation of state $$ (P + n^2 a / V)(V - n b) = nRT $$ it is easy to see that, for fixed $P$, $V$, $T$, the number of moles $n$ depends on the parameters $a$ and $b$, which in general will change from one type of gas to the other.

"My mental image of this is that it would be like adding water to a bucket of marbles, and that, evidently, is wrong, but why is it wrong?"

To the extent that a gas can be represented by the "ideal gas" model, the volume of the gas molecules is negligible compared to the volume of the container. Your intuition would apply in the limit of very high densities.

For the ideal gas case, you might think of the volume as being supported by pressure in a manner similar to how popping popcorn can support a volume in excess of the volume of the popcorn itself. It happens dynamically; the volume is almost entirely empty of molecules.

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Avogadro's hypothesis.

• noun the principle that equal volumes of all gases (given the same temperature and pressure) contain equal numbers of molecules synonyms: Avogadro's law see more see less type of: law , law of nature a generalization that describes recurring facts or events in nature

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