**Maxwell’s Distribution of ****Velocities and Energies **

**Maxwell’s Distribution of Molecular Velocities**

**Actually, as a result of ****collisions****, a **__redistribution__** of ****both velocity and energy ****takes place.**

**• But**** ****for convenience, **__i____t was supposed__** that** *all molecules in a given gas at a given temperature were moving with a*** ****constant root-mean-square velocity u****.**

**By utilizing probability considerations**, Maxwell and Boltzmann, have, in fact, shown that

• the **actual distribution of molecular velocities** depends on the temperature and molecular weight of a gas, and is given by the equation :

• This equation is known as the **Maxwell-Boltzmann distribution law for molecular velocities.**

**where, **

**• dn = the ****number of molecules out of a**** total molecules n having velocities between ****c and c+dc, **

**• M = molecular weight of the gas &**

**• T = temperature of the gas.**

where **p** denotes the fraction of molecules

**Fig.1**: **Distribution of Molecular Velocities**

**It is clear from the above plots**** of ****p ****vs**** c ****that **

• the probability of a molecule with zero velocity is very small.

• Further, the **probability of molecules with velocities greater than zero** increases, passes through a maximum, and then falls rapidly towards zero.

**The important features are:**

1. The fraction of molecules **with too low or too high velocities** is very small,

**2. M****ost probable velocity**➠There is a certain velocity for which the **fraction of molecules is maximum** viz., called **most probable velocity. **

• Thus **it is the velocity which is possessed by maximum number of molecules of the gas at a given temperature. **

• The most probable velocity is in any gas **not a constant**, **but shifts towards higher values of c with increase in temperature**; i.e., at higher temperatures higher velocities are more probable than at low.

**• Mathematical analysis shows that **the **most probable velocity, ****𝛼(****alpha****)**, __is not equal__** **either to the root- mean-square velocity u **or** the average velocity of all the molecules v.

**Effect of Temperature on Maxwell’s distribution of velocities : **

**It is clear from the curves of Fig.1 that **

__• O____n____ increasing the temperature__** ,****the peak of the distribution curve shifts ****downwards**** and the ****curve is flattened****.**

__• O____n____ increasing the temperature,__**t****he entire curve shifts more and more towards the ****right****.**

• __O____n____ increasing the temperature,__** ****t****he most probable velocity (****𝛼****)****increases****.**

__• O____n____ increasing the temperature__**,** **the fraction of molecules possessing the most probable velocity (****𝛼****) ****decreases****.**

__• At higher temperatures__** ****higher velocities are ****more probable**** than at low temperature.**

**Maxwell’s distribution of energies**

**According to Maxwell’s distribution of velocities:**

**As we know,**** **for any specific velocity c **the kinetic energy of a gas per mole is ****E = M c² / 2**

**substituting this relation**** in the above equation,**** ****the distribution of kinetic energies in a gas is obtained as:**

**where,**

** **Now **dividing** above equation by **dE, we get:**

• where **p’** =**probability for incidence of kinetic energies of translation of magnitude E.**

** • Note-** The

**plots of**

**p’ vs E**

**obtained from this equation are**

**very similar**to those of

**p vs c**as shown in

**Fig.1**