Measurement Size of Atom Avogadro

Introduction

Various scientists are making several attempts to measure the size of atoms. But Avogadro's hypothesis gives an important explanation for it. Avogadro's hypothesis was accepted after two years of Avogadro's death. It was accepted when Italian chemist Stanislao Cannizzaro was able to explain chemical exceptions to Avogadro's hypothesis. He explained and proved the relationship between atoms and molecules and the various weights of the materials.

Now, we will discuss the Avogadro hypothesis and learn how to calculate the size of atom using Avogadro law.

What was Avogadro?

Amedeo Avogadro or full name Lorenzo Romano Amedeo Carlo Avogadro (Born on 9 August 1776 and died 9 July 1856) was an Italian scientist. He was well known for his investigation of gas Volume(V), Pressure(P), and Temperature(T). He derived the gas law that is known as Avogadro's law or Avogadro hypothesis. For the study of atomic theory, Avogadro is regarded as an early figure and is an important contributor today.

He studied ecclesiastical law and practiced on his own. After that Avogadro began with his private study in two subjects such as physics and mathematics. With his brother, he conducted his first experiment on the subject of physics.

After that, he started teaching where he used to experiment with gas densities. He became the first chair of mathematical physics at the University of Turin. His major contribution was he clarified the confusion between atoms and molecules. In honor of Avogadro, the number of molecules in one mole is termed to be Avogadro number which is also known as Avogadro's constant. The experimental value of molecules per mole gram is $\mathrm{6.023\:\times10^{23}}$

Avogadro hypothesis

The Avogadro hypothesis is also known as the Avogadro principal or Avogadro law.

It states that at constant temperature and pressure, the total number of atoms or molecules of the gas is directly proportional to the volume occupied by any gas.

Now if we take two ideal gases and are mixed in equal quantities, then they contain an equal number of molecules. It is possible if the gas(which shows ideal behavior) is kept at the same pressure and temperature. So mathematically we can write,

$$\mathrm{V\:\varpropto\:n}$$

or we can write,

$$\mathrm{\frac{V}{n}\:=\:k}$$

Where, V - Volume of gas

n - gaseous substance, and K - constant for particular pressure and temperature In different conditions, if compared the same gaseous substance

$$\mathrm{\frac{V_{1}}{n_{1}}\:=\frac{V_{2}}{n_{2}}}$$

So from the above equation, we determine as an increase in gas volume then the amount of moles in the gas also increases. Same as in decreasing cases also. Therefore in a particular volume of gas, the total number of molecules or atoms present is independent of the molar mass of gas.

Avogadro Number

Avogadro’s number is defined as, in one mole of the substance which is determined as its molecular weight measured in grams, the number of units is equal to $\mathrm{6.023\:\times10^{23}}$

The consideration is the same for light as well as heavy gas. The unit we discussed may be an electron, ion, atom, or molecule which depends on the nature of the material and any disposition of the reaction.

How the Avogadro hypothesis helps to find the size of an atom?

By using Avogadro's hypothesis we can figure out the radius of an atom. Let in a substance molar mass ‘M’ and number of atoms N,

$$\mathrm{Number\:of\:atoms\:in\:a\:gram\:=\:\frac{N}{M}}$$

$$\mathrm{The\:total\:volume\:of\:atoms\:in\:a\:gram\:=\:\frac{N}{M}\:\times\frac{4}{3}\:\Pi\:r^{3}}$$

From Avogadro’s hypothesis, the actual volume of an atom in a gram of substance is two-thirds the volume of a gram of the substance.

$$\mathrm{\:\frac{N}{M}\:\times\frac{4}{3}\:\Pi\:r^{3}\:=\:\frac{2}{3}v}$$

{{As we know 𝐷𝑒𝑛𝑠𝑖𝑡𝑦, $\mathrm{\rho\:=\:\frac{Mass}{Volume}\:=\:\frac{1}{v}}$}

Thus, $\mathrm{\:\frac{N}{M}\:\times\frac{4}{3}\:\Pi\:r^{3}\:=\:\frac{2}{3\rho}}$

$$\mathrm{r^{3}\:=\:\frac{2\:\times\:3M}{3\rho\pi4N}\:=\:\frac{M}{2\pi\:N\rho}}$$

$$\mathrm{r\:=[\:\frac{M}{2\pi\:N\rho\:}]^{\frac{1}{3}}}$$

So from this formula, we can easily calculate the radius of the atoms and thus calculate the size of an atom.

Solved Examples -

Example1: 20 L of volume occupying 10 moles of air in a flywheel. But sometimes the flywheel loses 50 L of volume. Find the amount of air puncture in the flywheel. (Take temperature and pressure kept constant)

Solution − Given, Initial amount of air, 𝑛1 $\mathrm{n_{1}\:=\:10\:mol}$

The initial volume of the flywheel, $\mathrm{v_{1}\:=\:20\:L}$

The final volume of the flywheel, $\mathrm{v_{2}\:=\:50\:L}$

As we know from Avogadro’s law

$$\mathrm{\frac{v_{1}}{n_{1}}\:=\:\frac{v_{2}}{n_{2}}}$$

$$\mathrm{n_{2}\:=\:\frac{v_{2}}{v_{1}}\:\times\:n_{1}}$$

$$\mathrm{n_{2}\:=\:\frac{50}{20}\:\times\:10}$$

$$\mathrm{n_{2}\:=25\:mol}$$

Therefore the amount of air puncture in the flywheel is 25 mol.

Example 2: Initially a balloon contains 2 moles of hydrogen gas filled and tends to 2.5 liters of volume. If 2 mole of hydrogen gas is added, then what will be the volume of the balloon? Taking into consideration the pressure and temperature be constant.

Solution − Given, Initial amount of gas $\mathrm{n_{1}\:=2\:mol}$

The initial volume of the balloon, $\mathrm{v_{1}\:=2.5\:L}$

The final amount of gas, $\mathrm{n_{2}\:=2\:+\:2\:=\:4\:mol}$

As we know from Avogadro’s law

$$\mathrm{v_{2}\:=\frac{V_{1}}{N_{1}}\:\times\:n_{2}}$$

$$\mathrm{v_{2}\:=\frac{2.5}{2}\:\times\:4\:=\:5\:liters}$$

So, new volume of the balloon is 5 liters.

Conclusion

The term molecule is derived from the word mole. Avogadro's law can determine how t gas amount (n) is related to its volume (v). It was found to be a direct relationship, which suggested that the gas volume is proportional to the number of moles of gas present in it. It suggests that there is 6.023 × 1023 molecules in one mole.

Avogadro wasn’t able to determine the hypotheses about diatomic molecular characteristics. With this hypothesis, we can conclude the chemical formula of various gas substances and the size of an atom based on combining the gas volumes at a certain temperature and pressure.

In this tutorial, we cover the Avogadro hypothesis, Avogadro law, measuring the size of atom, and the limitation of Avogadro hypothesis.

FAQs

1. Name the term that describes the 𝟔. 𝟎𝟐𝟑 × 𝟏𝟎𝟐𝟑 characteristic particles?

Molar mass is the term that describes the 6.023 × 1023 characteristic particles

2. Write some applications of Avogadro's law?

Some applications of Avogadro’s law are −

• It helps to find the atomicity of gases.

• Gives a relation between molecular mass and vapor density.

• Explicate the gay lussac law.

• It helps in the molecular formula of gas.

3. What will be the pressure if the volume of a given gas at constant temperature converts 3 times. If ‘p’ is the initial pressure of the gas?

According to Boyle’s law

$$\mathrm{P_{1}\:V_{1}\:=\:P_{2}\:V_{2}}$$

$$\mathrm{\rho\:V_{1}\:=\:P_{2}\:3V_{1}}$$

$$\mathrm{P_{2}\:=\:\frac{\rho}{3}}$$

Therefore the pressure becomes,$\mathrm{\frac{\rho}{3}}$

4. What is the limitation of Avogadro law?

It is only applicable in lighter molecules(like hydrogen or helium). As it is applies only to real gas. This leads to the limitation of this law despite of applying to an ideal gas.

5. What are the other names of gas constants?

A gas constant is also known as an ideal gas constant, the universal gas constant, or the molar gas constant.