Kinetic Theory of Gases

The Kinetic Theory of Gases explains gas behavior by focusing on the random motion of molecules, which collide with each other and the container walls, creating pressure.The theory explains the larger properties of gases, such as pressure, volume, and temperature, by examining what happens at the molecular level. It also serves as the basis for understanding ideal gases, leading to important concepts like molecular speeds, temperature, and degrees of freedom, and is key to deriving the ideal gas law.

1.0Kinetic Energy of Gas

Gas molecules move randomly, performing translational, rotational, and vibrational motions, each contributing to their kinetic energy. The total kinetic energy of a gas depends on its temperature; higher temperatures increase the kinetic energy of the molecules.

2.0Potential Energy

• Intermolecular attraction force exists between molecules and thus every molecule has some potential energy.
• Greater Intermolecular force (IMF), lesser will be potential energy and greater will be magnitude of potential energy.

Orders

(1) IMF(Intermolecular Force)

Solid > Liquid > Gas (Real) > Ideal Gas (IMF = 0)

(2) Potential Energy

Solid < Liquid < Gas (Real) < Ideal Gas (IMF = 0)

3.0Internal Energy of Gas

Internal energy of gas is defined as some of potential energy and kinetic energy at a temperature.

At a given temperature for solid, liquid and gas:

1. Internal kinetic energy : Same for all
2. Internal potential Energy : Maximum for ideal gas (PE = 0) and Minimum for solids (PE = –ve)
3. Internal Energy : Maximum for Ideal gas and Minimum for solid

4.0Postulates of Kinetic Theory of Gases

• Every gas is made up of tiny particles called molecules. The molecules of a specific gas are identical to each other, but they differ from the molecules of other gases.
• The size is negligible in comparison to inter molecular distance10-9m.
• The gas molecules keep colliding among themselves as well as with the walls of the containing vessel. These collisions are perfectly elastic.
• There are no attractive or repulsive forces acting between the molecules of a gas.
• Gravitational attraction between the molecules is negligible because of their tiny masses and very high speeds.
• Molecules continuously collide with the walls of the container, resulting in changes to their momentum. This change in momentum is transferred to the walls, creating pressure exerted by the gas molecules on the container.

5.0Ideal Gas Equation

A gas which follows all gas laws and gas equations at every possible temperature and pressure is known as ideal or perfect gas.

$PV=\mu RT$

$\mu$ =Number of moles of gas

$PV=\frac{M}{M_w}RT$

$PV=\frac{mN}{m{N}_0}RT$

$PV=\frac{R}{N_0}NT$

$PV=kNT$         Boltzman Constant,$k=\frac{R}{N_0}$

$PV=\frac{\rho RT}{M_w}$             ($\rho =\frac{M}{V}$)

6.0Properties of Ideal Gas

• Ideal gas molecules can do only translational motion, so their kinetic energy is only translational kinetic energy.
• Ideal gas can not be liquified because the IMF is zero.
• The specific heat of ideal gas is constant quantity and it does not change with temperature.
• All real gases behave as an ideal gas at high temperature and low pressure and low density.
• Gas molecules have point mass and negligible volume and velocity is very high

 (107 cm/s). That's why there is no effect of gravity on them.

7.0NTP & STP

1atm=$1.01325\times 10^5$ Pascal

$1Litre=10^{-3}{m}^3=10^{3}c{m}^3=10^{3}cc=10^{3}ml$

8.0Gas Laws

1. Charle's Law: This law states that, for a fixed amount of an ideal gas at fixed pressure, the volume is directly proportional to its absolute temperature,

V ∝ T   m and P are Constant

$\frac{V_1}{V_2}=\frac{T_1}{T_2}$

Graphs for Charles's Law

Here t represents temperature in °C and T represents in K i.e. absolute temperature.

2. Boyle's Law : This law states that, for a fixed amount of an ideal gas at fixed temperature, the volume is inversely proportional to its pressure.

$V\propto \frac{1}{P}$ (m and T are constant)

$P_1{V}_1=P_2{V}_2$

Graphs for Boyle’s Law

3. Gay–Lussac's Law: According to this law, for a given mass of an ideal gas; at fixed volume, pressure of a gas is directly proportional to its absolute temperature,

P ∝ T    m and V are Constant

$\frac{P_1}{P_2}=\frac{T_1}{T_2}$

Graph for Gay–Lussac's Law

Here t represents temperature in °C and T represents in K i.e. absolute temperature.

4. Avogadro's Law: This law states that, at the same temperature and pressure, equal volumes of all gases contain an equal number of molecules, i.e., $N_1=N_2$ if P, V and T are the same. 
5. Dalton's Partial Pressure Mixture Law : This law states that the pressure exerted by a mixture of non-reactive gases is equal to the sum of the partial pressures of each gas in the mixture. $P=P_1+P_2+P_3+..................$

9.0Different Types of Speeds of Gas Molecules

1. Average velocity: Because molecules are in random motion in all possible directions with all possible velocity. Therefore, the average velocity of the gas molecules in the container is zero.

$<\vec{v}>\frac{\overrightarrow{v_1}+\overrightarrow{v_2}+...+\overrightarrow{v_n}}{N}=0$

2. Average speed or Mean speed $(v_{avg})$: By Maxwell’s velocity distribution law 

$<\vec{v}>\equiv v_{mean}\equiv v_{avg}=\frac{|\overrightarrow{v_1}|+|\overrightarrow{v_2}|+....+|\overrightarrow{v_n}|}{N}$

3. Most probable speed $(v_{mp})$: At a given temperature, the speed to which maximum number of molecules belongs is called as most probable speed $(v_{mp})$
4. RMS Speed $(v_{rms})$: RMS Speed (root mean squared speed) is equal to square root of the sum of square of speed of each molecule of a gas.

$v_{rms}=\sqrt{\frac{|\overrightarrow{v_1}|+|\overrightarrow{v_2}|+....+|\overrightarrow{v_n}|}{N}}$

10.0Maxwell's law of distribution of velocities

Maxwell had drawn a curve between number of molecules, moving with a particular velocity, and velocity of molecules known as velocity distribution curve shown here.

According to this law

• The velocities of molecules of a gas are in between zero and infinity (0 –∞ ).
• The area of the graph with the velocity axis gives the total number of molecules available in the container which remains constant.
• With the increase in the temperature, the most probable velocity increases but corresponding number of molecules decreases.
• The number of molecules within a certain velocity range is constant although the velocity of the molecule changes continuously at particular temperature.
• According to Maxwell $v_{rms}>v_{mean}>v_{mp}$

11.0Expression for Pressure of An Ideal gas

$P=\frac{1}{3}\frac{M}{V}{v}_{rms}^2=\frac{1}{3}\rho v_{rms}^2$

All gas laws and gas equation can be obtained by expression of pressure of gas (except Joule’s law)

12.0Translational Kinetic Energy

$E_T=\frac{1}{2}M{V}_{rms}^2$

$E_T=\frac{1}{2}M\frac{3P}{\rho }$

$E_T=\frac{3}{2}PV$

$\frac{E_T}{V}=E_v=EnergyDensity$

$E_v=\frac{3}{2}P$

13.0Expression of RMS Speed

$P=\frac{1}{3}\rho (V_{rms}{)}^2$

$V_{rms}=\sqrt{\frac{3P}{\rho }}$

$\frac{P}{\rho }=\frac{RT}{M_w}=\frac{K_{b}T}{m}$

$V_{rms}=\sqrt{\frac{3P}{\rho }}=\sqrt{\frac{3RT}{M_W}}=\sqrt{\frac{3k_{b}T}{m}}$

If gas is same (Mw=const..) and temp. is changed

$V_{rms}\propto \sqrt{T}$

If temperature is constant and gas is changed

$V_{rms}\propto \frac{1}{\sqrt{M_W}}$

14.0Effect of Pressure at Constant Temperature

$V_{rms}\propto \frac{1}{\sqrt{M_W}}$

$P=\frac{\rho RT}{M_w}$

T & Mw=Constant

$\frac{P}{\rho }=Constant$

So no effect on $V_{rms}$

Note:

1. If only temp. is changed

$V_{rms}=\sqrt{\frac{3RT}{M_w}}$

2. If only pressure is changed

$V_{rms}=\sqrt{\frac{3P}{\rho }}$

3. If both pressure & temp. is changed

$V_{rms}=\sqrt{\frac{3k_{b}T}{M_w}}$

Most probable speed

$v_{mp}=\sqrt{\frac{2P}{\rho }}=\sqrt{\frac{2RT}{M_w}}=\sqrt{\frac{2k_{b}T}{m}}$

Average speed

$v_{avg}=\sqrt{\frac{8P}{\pi \rho }}=\sqrt{\frac{8RT}{\pi M_w}}=\sqrt{\frac{8k_{b}T}{\pi m}}$

RMS Speed

$v_{rms}=\sqrt{\frac{3P}{\rho }}=\sqrt{\frac{3RT}{M_w}}=\sqrt{\frac{3k_{b}T}{M_W}}$      

$v_{rms}>v_{mean}>v_{mp}$

15.0Escape Velocity

The minimum velocity required for which molecule or object to escape out from the earth surface.           

$V_{es}=\sqrt{\frac{2GM}{R}}$

$(V_{es}{)}_{Earth}=11.2km/sec$

16.0Relation of Escape Velocity With RMS

Condition for atmosphere to be present at the surface of any planet:

$V_{rms}<V_{es}$→ Atmosphere present

$V_{rms}\ge V_{es}$→ No atmosphere

$\sqrt{\frac{3RT}{M_w}}\ge V_{es}$

$T\ge \frac{(V_{es}{)}^2{M}_w}{3R}$

$T_{min}\ge \frac{(V_{es}{)}^2{M}_w}{3R}$

$T_{min}=\frac{M_w}{2}\times 10^{7}K\to here{M}_{w}isinKg$

$T_{min}=\frac{M_w}{2}\times 10^{4}K\to here{M}_{w}isingram$

Note: If we continuously increase the temperature at $10^4$ K hydrogen gas molecules will leave the earth surface first. The moon has no atmosphere because the R.M.S  speed of gas molecules is greater than the escape velocity.

17.0Degrees of Freedom $(f)$

• The number of independent ways in which a molecule or an atom can exhibit motion or have energy is called its degrees of freedom.

                                                  Or

• The number of independent coordinates required to specify the dynamic state of a system is called its degrees of freedom.

(a)Translational Degree of freedom : Number of independent ways in which a gas molecule may perform translatory motion. There are a maximum of three degrees of freedom corresponding to translational motion.

(b)Rotational Degree of freedom : Number of independent ways in which a gas molecule may perform rotational motion. The number of degrees of freedom in this case depends on the structure of the molecule.

(c) Vibrational Degree of freedom : Number of independent ways in which a gas molecule may vibrate. It is exhibited at high temperatures only.

Note:

Degrees of freedom depend on atomicity, structure & temperature of gases.

Total number of degree of freedom = 3N

Where N = Atomicity of gas molecule

For a gas molecule

Vibrational DoF = Total DoF – (Translational + Rotational) DoF

18.0Law of equipartition of energy

The total kinetic energy of gas molecules is evenly distributed across all degrees of freedom, and the energy corresponding to each degree of freedom at absolute temperature T  is $\frac{1}{2}kT$

For one molecule of Gas

Energy related with each degree of freedom =$\frac{1}{2}kT$

Energy related with all degree of freedom=$\frac{f}{2}kT$

$\overline{v_x^2}=\overline{v_y^2}=\overline{v_z^2}=\frac{v_{rms}^2}{3}$

$\Rightarrow \frac{1}{2}m{v}_{rms}^2=\frac{3}{2}kT$

So energy related with one degree of freedom$=\frac{1}{2}m\frac{v_{rms}^2}{3}=\frac{3}{2}\frac{kT}{3}=\frac{1}{2}kT$

Note: For ideal gas if atomicity is not known then take f = 3.

19.0Specific Heat

The amount of energy required to elevate the temperature of the unit mass of a substance by 1°C (or 1K) is called its specific heat. It is represented by s or c.

$c=\frac{1}{m}\frac{dQ}{dt}$

1. Gram specific heat :The amount of energy required to elevate the temperature of 1 gram of a substance by 1°C (or 1K) is called its specific heat.

$c_{gram}=\frac{1}{m}\frac{Q}{\Delta t}$

Unit: joule/kg-°C,cal/g-°C

2. Molar specific heat :The amount of energy required to elevate the temperature of 1 mole of a substance by 1°C (or 1K) is called its specific heat.

$c_{molar}=\frac{1}{\mu }\frac{Q}{\Delta t}$

Unit: joule/mol-°C,cal/mol-°C

Note 

•  The value of specific heats of gas can vary from zero (0) to infinity.
• Generally two types of specific heats are defined for a gas –

                 (a) Specific heat at constant volume $(c_V)$

                 (b) Specific heat at constant pressure $(c_P)$

• There are many possible processes to give heat to a gas.

A specific heat can be associated with each such process which depends on the nature of the process.

1. Isochoric Process

      V = Constant

       $c_V=\frac{1}{\mu }\frac{Q}{\Delta t}$

2. Isobaric Process

       P= Constant

       $c_P=\frac{1}{\mu }\frac{Q}{\Delta t}$

3. Isothermal Process

       T=Constant 

       $c_{Isothermal}=\infty$

4. Adiabatic Process

         Q=0

         $c_{Adiabatic}=0$

20.0Mayer’s Law

For an ideal gas

$c_V=\frac{f}{2}R$

$c_V=(\frac{f}{2}+1)R$

$c_P-c_V=R$

If gram specific heat are given

$(c_P{)}_{gram}-(c_V{)}_{gram}=\frac{R}{M_w}$

Molar Specific Heat For Mixture of Gases

$C_{mix}=\frac{{\mu }_1{c}_1+{\mu }_2{c}_2}{{\mu }_1+{\mu }_2}$

$(C_v{)}_{mix}$$=\frac{{\mu }_1{c}_{v1}+{\mu }_2{c}_{v2}}{{\mu }_1+{\mu }_2}$ $(\because {\mu }_{mix}={\mu }_1+{\mu }_2)$

$(C_P{)}_{mix}$$=\frac{{\mu }_1{c}_{p1}+{\mu }_2{c}_{p2}}{{\mu }_1+{\mu }_2}$

${\gamma }_{mix}$$=\frac{(c_p{)}_{mix}}{(c_v{)}_{mix}}$$=\frac{{\mu }_1{c}_{p1}+{\mu }_2{c}_{p2}}{{\mu }_1{c}_{v1}+{\mu }_2{c}_{v2}}$

21.0Mixture of Gases

Case – I :Here two gases kept in two different vessels are mixed in another vessel.

$\frac{P_1{V}_1}{T_1}+\frac{P_2}{V_2}{T}_2=\frac{P_3{V}_3}{3}$

Case – II: Here two gases are kept in two different vessels connected via a tube as shown. Cross-section area of this tube is very small so its volume can be neglected w.r.t. volume of vessel.

$\frac{P_1{V}_1}{T_1}+\frac{P_2{V}_2}{T_2}=\frac{P{V}_1}{T_1}+\frac{P{V}_2}{T_2}$

22.0Mean Free Path and Real Gases

The mean distance travelled by a molecule among two successive collisions is called the mean free path m of the molecule.${\lambda }_m$

${\lambda }_m=\frac{1}{\sqrt{2}{d}^{2}n}$

$n=\frac{N}{V}$=No. of molecules per unit volume

d=Diameter of a Molecule

Real Gas

Practically, no gas is ideal at all. They show some deviation from ideal gas behaviour and are known as Real gas.

Main cause of deviation is two postulates that have been assumed in Kinetic Theory of Gases :-

(1) Interaction force between gas molecules are considered negligible

(2) Size of Gas molecules is considered negligible. But practically they are not negligible.

23.0Compressibility Factor

It is used to indicate the degree to which a real gas deviates from ideal gas behavior. It is denoted by Z.

$Z=\frac{P{V}_{Real}}{\mu RT}$

$Z=\frac{P{V}_{Real}}{P{V}_{Ideal}}=\frac{V_{real}}{V_{ideal}}$

Case-1

Z = 1

$V_{real}=V_{ideal}$

Thus real gas behaves as ideal gas.

Case-2

Z > 1

$V_{real}>V_{ideal}$

Thus real gas expands with respect to ideal gas and hence we can say positive deviation.

Case-3

Z <1

$V_{real}<V_{ideal}$

Thus real gas compresses with respect to ideal gas and hence we can say negative deviation.

24.0Compressibility Factor Curve

• Thus, we can see from the curve that real gas approaches ideal gas behaviour at Low Pressure and High Temperature.

25.0Van der Waal's Equation for Real Gas

1. Pressure Correction: Since we have neglected intermolecular interaction, thus actual pressure of gas will be less than ideal pressure and it was found that

                    $P_{ideal}=P_{real}+\frac{{\mu }^{2}a}{V^2}$

                     a=Constant and value depends on real gas

2. Volume Correction: Also we have neglected size of molecules but actual volume occupied by real gas will be

          $V_{ideal}=V_{real}-\mu b$

             b = Constant and value depends on real gas

            $P_{ideal}{V}_{ideal}=\mu RT$

            $(P_l+\frac{{\mu }^{2}a}{V^2})(V-\mu b)=\mu RT$

This equation is known as Van der Waal's Equation for Real Gas.

26.0Sample Questions On Kinetic Theory of Gases

Q-1. What is Mean Free Path?

Solution: The average distance travelled by a molecule between two successive collisions is called the mean free path ${\lambda }_m$ of the molecule.

${\lambda }_m=\frac{1}{\sqrt{2}\pi d^{2}n}$

$n=\frac{N}{V}=$No. of molecules per unit volume

d=Diameter of a Molecule

Q-2. If 4 moles of a monoatomic gas and 2 moles of a diatomic gas are mixed then find value of  $\gamma$ for the mixture.

Solution:

Monoatomic gas ${\mu }_1=4,C_{v1}=\frac{3}{2}R,C_{P1}=\frac{5}{2}R$

Diatomic Gas ${\mu }_2=2,C_{v2}=\frac{5}{2}R,C_{P2}=\frac{7}{2}R$

${\gamma }_{mix}$$=\frac{(C_p{)}_{mix}}{(C_v{)}_{mix}}$$=\frac{{\mu }_1{c}_{p1}+{\mu }_2{c}_{p2}}{{\mu }_1{c}_{v1}+{\mu }_2{c}_{v2}}$$=\frac{4\times \frac{5}{2}R+2\times \frac{7}{2}}{4\times \frac{3}{2}R+2\times \frac{5}{2}R}$$=1.54$

Q-3. If 4 moles of diatomic gas is heated at constant volume by giving 300R heat. Then calculate the change in temperature of the gas.

Solution: At constant volume

$C_v=\frac{5}{2}R$   (for diatomic gas)

$\mu c_V\Delta T=Q$

$4\times \frac{5}{2}R\times \Delta T=300R$

$\Delta T=30K$

Q-4. At a constant temperature if the pressure of a gas is tripled. Find fractional change in most probable speed.

Solution:

$v_{mp}=\sqrt{\frac{2P}{\rho }}=\sqrt{\frac{2RT}{M_w}}$

At constant temperature $\frac{P}{\rho }$ is constant hence $v_{mp}$ does not change by changing pressure.