Kinetic Theory of an Ideal Gas: Statement, Proof, Examples

Kinetic Theory of an Ideal Gas: We know that everything in the universe is made up of atoms and molecules. From the hardest substance to the air that surrounds us, everything is made up of atoms and molecules. Therefore, the properties we observe at the macroscopic level can also be explained at the atomic or molecular level. Kinetic theory is one such attempt to explain the behaviour of gases using the properties at the microscopic level. Let us read further to know more about this topic.

Molecular Nature of Matter

Everything in this universe is made up of atoms and molecules. Even before John Dalton’s introduction of the atomic viewpoint in modern science, scholar’s in ancient India, like Kannada, stated that matter is not continuous. In the Vaiseshika school of thought, founded by Rishi Kannada in \({\rm{600}}\,{\rm{BC,}}\) the atomic theory was discussed thoroughly. It was set that if any two different substance (like an iron and a wood) is divided endlessly, we will reach a point where the constituent particle obtained will be the same.

Learn Kinetic Theory of Gases

In the modern-day, John Dalton is credited with the introduction of the atomic viewpoint of the matter.

Dalton’s atomic theory proposed that all matter present in the universe is composed of atoms that are indivisible and indestructible building blocks. While all atoms of an element were identical, different elements had atoms of different size and mass

What is the Kinetic Theory of Ideal Gases?

In solids, the constituent particles(like atoms in molecules) cannot movely, but in the case of fluids(liquid and gas), the atoms or molecules are to move. In the case of gases, the intermolecular forces are so small that the molecules can move, neglecting the gravity. Therefore a gas captain a container exerts pressure on the walls of the container. this pressure is actually due to the collision of the molecules of the gas from the walls of the container

James Clark Maxwell, Clausius, and Rudolph proposed the basic postulates of the kinetic theory of gases.

1. Gases consist of a large number of extremely small particles, i.e., atoms and molecules. The size of the particles is negligible as compared to the distance between any two particles

2. The volume of the gaseous particles is negligible. This is because we consider the size of the particle to be negligible.

3. These particles are in random motion that results in collisions with each other and with the walls of the container. Pressure is exerted by the gas molecules due to such collisions. 

4. The collision among the particles and between the particles on the walls of the container is perfectly elastic; therefore, no kinetic energy is lost due to the collisions
5. The average kinetic energy of the particles signifies the absolute temperature of the gas. the average kinetic energy is directly proportional to the absolute temperature
6. Intermolecular forces between any two particles or molecules are zero.

Terms Related to the Kinetic Theory of Gases

Mean Path: The average distance a molecule can travel without colliding is called the mean path. The mean path is larger in gases as they can move morely in solids. The mean path is very small as the molecules are closely packed.

Root Mean Square (RMS) velocity: It is the square root of the average of the square of the velocities of all the molecules present.

\({V_{rms}} = \sqrt {\frac{{3RT}}{M}} \)

Where\(:R\) is the gas constant

\(T\) is the absolute temperature

\(M\) is the molar mass

Most Probable Velocity: It is the velocity possessed by the maximum number of gas molecules in a sample.

\({V_{mp}} = \sqrt {\frac{{2RT}}{M}} \)

Where\(: R\) is the gas constant

\(T\) is the absolute temperature

\(M\) is the molar mass

Average speed: It is the average of the velocity possessed by molecules of the gas.

\({V_{av}} = \sqrt {\frac{{8RT}}{{\pi M}}} \)

Where\(: R\) is the gas constant

\(T\) is the absolute temperature

\(M\) is the molar mass

Boltzmann-constant: 

\(k = n\frac{R}{N}\)

Where\(: R\) is the gas constant

\(n\) is the number of moles

\(N\) is the number of particles in one mole (the Avogadro number)

Total translational Kinetic Energy of gas: Since all the intermolecular interactions are neglected, the total internal energy will only be in the form of kinetic energy.

\(K.E. = \frac{3}{2}nRT\)

Where,

\(n\) is the number of moles

\(R\) is the universal gas constant

\(T\) is the absolute temperature

Avogadro’s law

At constant pressure, the amount (moles) of gas is directly proportional to the volume of the gas.

\(\frac{{\rm{V}}}{{\rm{N}}}{\rm{ = Constant}}\)

Boyle’s law

At constant temperature, The pressure is inversely proportional to the volume of the gas.

\(PV = {\rm{Constant}}\)

Charles’s law

At constant pressure, the volume of the gas is directly proportional to the temperature of the gas.

\(VT = {\rm{Constant}}\)

Pressure of an Ideal Gas

Let us consider a container containing gaseous molecules.

According to Newton’s second law of motion, the force on the walls of the container is equal to the change in momentum of the gaseous molecules per unit time after the collision.

Thus, \(F = \frac{{\Delta p}}{{\Delta t}} = \frac{{m\Delta v}}{{\Delta t}}\)

Where

 \(p\) denotes momentum.

\(m\) denotes the mass of the particle.

 \(Δt\)  denotes the time interval.

Since the collision is elastic, the gaseous molecule rebounds at the same speed after colliding with the wall of the container.

\(\Delta p = m{V_x} – \left( { – m{V_x}} \right) = 2m{V_x}\)

\( \Rightarrow F = \frac{{\Delta p}}{{\Delta t}} = \frac{{2m{V_x}}}{{\Delta t}}\)

Now, assume the length of the sides of the box as ‘\(L\)’. Time interval after which the same gas molecule will again collide with the wall will be,

\({V_x} = \frac{{2L}}{{\Delta t}}\)

Thus, \(\Delta t = 2L/Vx\)

Hence, substituting the value of \(Δt,\) we get,

\(F = \frac{{mV_x^2}}{L}\)

Total force will be the sum of all forces due to all the molecules.

\(F = \frac{{m\left( {V_{x1}^2 + V_{x2}^2 + \ldots V_{xn}^2} \right)}}{L}\)

Dividing L.H.S. and R.H.S by \(N\) (number of particles):

\(\frac{F}{N} = \frac{{m\left( {V_{x1}^2 + V_{x2}^2 + \ldots V_{xn}^2} \right)}}{{LN}}\)

Therefore,  \(\frac{F}{N} = \frac{m}{L}\overline {V_x^2} \)

\(F = N\frac{m}{L}\overline {V_x^2} \)

Dividing L.H.S. and R.H.S by A (area):

\(\frac{F}{A} = N\frac{m}{{A.L}}\overline {V_x^2} \)

We know that Area is \({L^2}\) and \(F/A\) is pressure. Therefore:

\(P = \frac{{NmV_x^2}}{{{L^3}}}\)

Now, we know that \({L^3} = V\) (volume) 

Thus, \(PV = Nm\overline {v_x^2} \)

If gas molecules move in all \(x, y,\) and \(z\) directions:

\({v_{{\rm{total }}}} = v_x^2 + v_y^2 + v_z^2\)

\(\left( {\overline {v_{{\rm{total }}}^2} } \right) = \left( {\overline {v_x^2} } \right) + \left( {\overline {v_y^2} } \right) + \left( {\overline {v_z^2} } \right)\)

Since particles can move randomly so we can generalize all the directions as \({v_x},\)Thus:

\(\left( {\overline {v_{{\rm{total }}}^2} } \right) = 3\left( {\overline {v_x^2} } \right)\)

Thus, \(\frac{{\left( {\overline {v_{{\rm{total }}}^2} } \right)}}{3} = \left( {\overline {v_x^2} } \right)\)

\(PV = \frac{1}{3}Nm\left( {v_{{\rm{total }}}^2} \right)\)

Maxwell Distribution Function

Every molecule has some speed which can vary with temperature, but at a constant temperature, the RMS velocity or the average velocity will be the same, but the distribution of the velocity among the molecules of the gas is determined by maxwell’s speed distribution.

\(P(v) = 4\pi {\left( {\frac{M}{{2\pi RT}}} \right)^{\frac{3}{2}}}{v^2}{e^{ – M{v^2}/2RT}}\)

Law of Equipartition of Energy and Degrees ofdom

Total Internal Energy: It is the property of a substance that signifies the total energy possessed by the constituent particles of that substance.

For gases, the internal energy consists of,

1. Kinetic Energy
   a. Translational Kinetic Energy
   b. Rotational Kinetic Energy
   c. Vibrational Kinetic Energy(Significant only at higher temperatures)

2. Potential energy
a. Intermolecular Potential Energy(it is zero for ideal gases)
b. Intramolecular Potential Energy(Significant only at higher temperatures)

Degrees ofdom: Degrees ofdom is the total number of possible independent motions for a given molecule.

According to the Law of Equipartition of Energy, any ideal gas distributes its total internal energy among its all degree ofdom equal.

The total internal energy of an ideal gas for its one mole and its one degree ofdom is given by,

\(\frac{1}{2}RT\)

\(T\) is the absolute temperature.

Thus, the total internal energy of the gas is given by,

\(U = \frac{1}{2}nfRT\)

Change in internal energy is given by,

\(\Delta U = \frac{1}{2}nfR\Delta T\)

\(n\) is the number of moles of the gas.

\(f\) is the total number of degrees ofdom.

\(R\) is the universal gas constant.

\(T\) is the absolute temperature.

Therefore, we can infer that the change in internal energy is a state function, i.e. it only depends on the final and the initial state of the gas.

Sample Problems

Q.1. A flask contains argon and chlorine gas in the ratio \(2:1\) by mass heart temperature \({\rm{300}}\,{\rm{Kelvin}}\) find the ratio of the average kinetic energy of the two gases. The atomic mass of argon is \({\rm{39}}{\rm{.9}}\,{\rm{u}}\) and the molecular mass of chlorine is \({\rm{70}}{\rm{.9}}\,{\rm{u}}\)
Ans: According to the kinetic theory of gases, the average kinetic energy of gas only depends on its absolute temperature and is independent of the nature of the gas.
Therefore the ratio of the kinetic energy of the two gases is \(1:1.\)

Q.2. In the above problem, determine the ratio of the root mean s square speed off the molecules after two gases.
Ans: The root mean square speed off the molecules of a gas is given by,
\({V_{rms}} = \sqrt {\frac{{3RT}}{M}} \)
Where\(: R\) is the gas constant
\(T\) is the absolute temperature
\(M\) is the molar mass
\(\frac{{{{\left( {{V_{rms}}} \right)}_{Ar}}}}{{{{\left( {{V_{rms}}} \right)}_{Cl}}}} = \sqrt {\frac{{{M_{Cl}}}}{{{M_{Ar}}}}} = \sqrt {\frac{{70.9}}{{39.9}}} = 1.33\)

Summary

The kinetic theory of gases explains the properties we observed on a macroscopic level using the principles at a microscopic level. The kinetic theory of gases takes ideal gas into consideration. The total energy of the gas is due to the kinetic energy possessed by the molecules, and it reflects the absolute temperature of the gas.

The different gas molecules pose different velocities, but the velocity that is possessed by the maximum number of molecules is known as the most probable velocity. We have two other kinds of velocity that is the average velocity and the root being square velocity. To Maxwell speed distribution curve reflects the velocities with reference to the number of molecules with that velocity.

We have different laws related to the behaviour of gases. We can, through any of these laws using the kinetic theory of gases. For ideal gases, the potential energy due to intramolecular forces is zero, but that is not the case with real gases. We can understand the behaviour of real gases by studying the kinetic theory for an ideal gas.

FAQs

Q.1. What is an ideal gas?
Ans: An ideal gas is a type of gas that does not possess any intermolecular forces. The size of the molecules of this gas is negligible.

Q.2. What is the significance of the total kinetic energy of the gas molecules?
Ans: The total kinetic energy signifies the absolute temperature of the gas. The total kinetic energy is directly proportional to the temperature of the gas.

Q.3. What is the most probable velocity?
Ans: The velocity that is possessed by the maximum number of the gaseous molecules is known as the most probable velocity. It is given by,
\({V_{mp}} = \sqrt {\frac{{2RT}}{M}} \)
Where\(: R\) is the gas constant
\(T\) is the absolute temperature
\(M\) is the molar mass

Q.4. What type of collision occurs between the walls of the container and the gaseous molecules and among the gaseous molecules themselves?
Ans: All the collision that is happening is perfectly elastic; therefore, there is no loss in the kinetic energy of the molecules.

Q.5. What is the degree ofdom?
Ans: The degree ofdom is the total number of possible independent motions for a given molecule.

We hope this detailed article on the kinetic theory of an ideal gas helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!