Kinetic Interpretation of Temperature and RMS Speed of Gas Molecules

In Kinetic theory, the kinetic interpretation of temperature states that the average kinetic energy of the gas molecules of an ideal gas is directly proportional to the absolute temperature of the molecules.

• The average kinetic energy is independent of the pressure, volume, and nature of the gas.
• The energy (per molecule) for any monatomic atoms like argon and diatomic molecules like chlorine is equal to (3/2) kBT.
• The kinetic interpretation of temperature relates to the internal energy and temperature of a molecule.
• It is determined that the average velocity of the molecules increases with the increase in the temperature of the gas.
• The kinetic molecular theory will explain the Boyle law, Charles law, Avogadro law and Dalton law.

Key Terms: Kinetic Interpretation of Temperature, RMS Speed of Gas Molecules, Average Speed of The Molecules of Gas, Kinetic energy, Internal energy, Ideal gas equation, Boltzmann constant, Gas Constant.

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Kinetic Theory of an Ideal Gas

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The kinetic theory of an ideal gas is based on the molecular picture of matter. According to which

• A given amount of gas is a mixture of a very large number of identical molecules of the order of Avogadro’s number.
• The molecules move randomly in all directions.
• At ordinary temperature and pressure, the size of the molecules is very small compared to the distance between them.
• Thus, the interaction between them is negligible. Hence they move according to Newton’s law of motion.
• Except during collisions, there is no force exerted between the molecules.
• The collisions of the molecules against each other or with the walls of the container are perfectly elastic.
• Therefore, though their velocities change, momentum and the kinetic energy of the molecules are conserved during the collision.

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Kinetic Interpretation of Temperature

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Kinetic Interpretation of Temperature is used to express relation between the temperature and the kinetic energy of the molecules. For gas molecules, as the temperature increases, the kinetic energy of the molecules also increases.

• The kinetic energy increases as molecules get more energy for movement.
• The results is similar in case of liquids.
• In the case of solids, with an increase in temperature, the molecules start vibrating with greater speed.

Derivation of Kinetic Interpretation of Temperature

Consider 1 mole of an ideal gas enclosed in a cube of volume V and the molecules of the gas are moving with velocity v, then the pressure exerted by the ideal gas on the wall of the container is given by

\(P=\frac{1}{3}mnv^2 \)

Where

• v = RMS (root mean square) velocity 
• P = pressure 
• m = mass of one molecule 
• n = N_A/V = number of molecules (N_A) per unit volume (V)
• N_A = Avagadro's number = number of molecules in one mole of gas

\(\Rightarrow P=\frac{1}{3}m\frac{N_A}{V}v^2 \)

\(\Rightarrow PV=\frac{1}{3}mN_Av^2\)   ...(i)

From the ideal gas equation, we have

PV = μRT 

Where

• μ = number of moles of the gas
• R = Gas constant
• T = Absolute temperature

For one mole, μ = 1

⇒ PV = RT

Substituting the above value in equation (i), we get

\( RT=\frac{1}{3}mN_Av^2\)

\(\)\(\Rightarrow \frac{RT}{N_A}=\frac{1}{3}mv^2\)

\(\Rightarrow \frac{RT}{N_A}=\frac{2}{3} \times \frac{1}{2} mv^2\)

\(\Rightarrow \frac {1}{2}mv^2= \frac {3}{2}\frac {RT}{N_A}\)    ...(ii)

Now the average kinetic energy of the molecules of the gas is given by

Kinetic energy,  

\((KE)_{avg}=\frac{1}{2}mv^2\)

Using equation (ii), we get

The average kinetic energy,

 \((KE)_{avg}= \frac {3}{2}\frac {RT}{N_A}\)

But, R = N_AK_B

Where

• K_B is the Boltzmann constant 
• N_A is Avagadro's number

Therefore, the average kinetic energy of the molecules of the gas is given by

\((KE)_{avg}= \frac {3}{2}\frac {N_AK_B}{N_A}T\) 

\(\Rightarrow (KE)_{avg}= \frac {3}{2}K_BT\)

The above equation shows that the average kinetic energy of a molecule of gas is directly proportional to the absolute temperature of the gas. This is the Kinetic interpretation of temperature. i.e.

\((KE)_{avg}=\frac {1}{2}mv^2= \frac {3}{2}K_BT\)

OR

\((KE)_{avg}\propto T\)

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Root Mean Square Speed (RMS Speed)

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Root mean square speed or RMS speed refers to the square root of the mean of the squares of the random speeds of the individual molecules of the gas. The speed will express the velocity of the gas molecules in terms of their temperature and molar mass.

• It is given by

\(v_{rms}=\sqrt{v^2}\)   ...(i)

• From the kinetic interpretation of temperature, the average kinetic energy of a molecule of gas is given by

\((KE)_{avg}=\frac {1}{2}mv^2= \frac {3}{2}K_BT\)

\(\Rightarrow v^2= \frac{3K_BT}{m}\)   ...(ii)

\(\Rightarrow v^2 \propto T\)

• Hence, using equation (i), we get

\(v_{rms} \propto \sqrt{T}\)

• i.e. rms velocity of the gas molecules is directly proportional to the square root of absolute temperature.
• Also, using equation (ii), we get

\( v_{rms}= \sqrt{\frac{3K_BT}{m}}\)

• But, K_B = R\N_A

\( \Rightarrow v_{rms}= \sqrt{\frac{3RT}{mN_A}}\)

Substituting mN_A = M, the molar mass of the gas, we get

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

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Things to Remember

• Kinetic interpretation of temperature is based on the assumption that the gas is an ideal gas. 
• The rms speed refers to the root mean square speed and it is calculated as the root of the average of the square of velocities.
• Root mean square speed is used to calculate the speed of particles that are in a constant state of motion.
• The ideal gas equation is PV = nRT.
• It is assumed that in the case of the Kinetic Theory of Gases, then, no force will be acting on the molecules except during elastic collisions.
• Temperature is directly proportional to the square of rms velocity; T ∝ v²

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