# Finding The Velocity Of A Gas

- Root Mean Square Velocity: We know that gases are made up of a large number of molecules. Now, the root mean square velocity of the gas is defined as the square root of the mean of the squares of the velocity of a large number of molecules present in the same gas. It is represented by the letter 'u'.
- Average (Mean) velocity: as denoted by the name it is the average of the velocity of all the molecules present within the gas. It is denoted by the letter v.
- Most Probable velocity: The velocity of the gas molecules keeps changing due to frequent collisions. However, certain portion of the gas molecules have the same velocity. Velocity possessed by a maximum number of gas molecules is termed as most probable velocity. It is denoted by α.

u= √[u

_{1}

^{2}+ u

_{2}

^{2}+ u

_{3}

^{2}-----------u

_{n}

^{2}/ n]

where u

_{1}, u

_{2}, u

_{3}are the velocities of the gas molecules.

n is the number of molecules in the gas.

Root mean square velocity or the RMS velocity is directly proportional to square root of absolute temperature and inversely proportional to the square root of molecular weight and density.

RMS velocity can also be derived from kinetic gas equation i.e. PV=RT= 1/3 M u

^{2}

u = √3RT/ M

= √3PV / M

= √3P / d (as M/V= d, density)

The value of R is given as 8.314 x 10

^{7}ergs K

^{-1}mole

^{-1}

Or 8.314 Joules K

^{-1}mole

^{-1}

Or 1.99 calories K

^{-1}mole

^{-1}

u

_{1}+ u

_{2}+ u

_{3}---------u

_{n}

v=-----------------------------------

n

where u

_{1}, u

_{2}, u

_{3}are the velocities of the gas molecules.

n is the number of molecules in the gas.

A relationship exists between the RMS velocity and the average velocity. It is given as

Average velocity (v) = 0.9213 x RMS velocity.

A = √(2RT / M)

= √(2RT/ N x m) ( as M = N x m)

RMS velocity can also be related to the most probable velocity by a relationship.

Relation between most propbale velocity (α) and RMS velocity is = √(2/3) x u.

= 0.816 x RMS velocity.

All the three velocities RMS, Average and the Most probable velocities can be related by the equation

α : v : u = 1: 1.128 : 1.234

**Solved problems**

- Velocity possessed by a maximum number of molecules in the gas is termed as
- Root mean square velocity
- Average velocity
- Most probable velocity
- Any of the above.

- Relationship between RMS velocity and the Average velocity is given by
- 0.9213 x RMS velocity
- 0.816 x RMS velocity
- They cannot be related
- Both are identical

- Relationship between RMS velocity and Most probable velocity is given by
- 0.9213 x RMS velocity
- 0.816 x RMS velocity
- They cannot be related
- Both are identical

- Calculate the RMS velocity, average velocity and most probable velocity of oxygen at 27
^{ο}C?

Root mean square velocity is given by the formula √3RT/M

So, R = 8.314 x 10^{7}ergs K^{-1}mole^{-1}

T= 273 + 27 = 300 K

M =32

Thus, u= √(3 x 0.8314 x 10^{7 }x 300/32)

= 1.58 x 10^{4}cm sec^{-1}

= 48200 cm sec^{-1}

Average velocity (v) = 0.9213 x RMS velocity.

= 0.9213 x 48200

=44406.66 cm/sec

Most Probable velocity = 0.816 x RMS velocity

= 0.816 x 48200

= 39331.2 cm/sec

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