Melting and Boiling Points, Molecular Speeds

1. Root mean square velocity (u_rms) : It is the square root of the mean of the squares of the velocity of a large number of moleucles of the same gas.
\tt u_{rms} = \sqrt{\frac{u_1^2 + u_2^2 + u_3^2 + .... u_n^2}{n}}
\tt u_{rms} = \sqrt{\frac{3PV}{\left(mN_{0}\right)}} = \sqrt{\frac{3RT}{\left(mN_{0}\right)}} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3P}{d}} 
Where, mN₀ = M
Where, k = Boltzmann constant = \tt \frac{R}{N_{0}}

2. When temperature alone is given then, \tt u_{rms} = 1.58 \ \times \sqrt{\frac{T}{M}} \times 10^{4} cm / sec

3. If P and T both are given, use equation in terms of temperature, i.e., \tt u_{rms} = \sqrt{\frac{3RT}{M}} and \ not \sqrt{\frac{3pV}{M}}

4. Average velocity (v_av) : It is the average of the various velocities possessed by the molecules.v_{av} = \frac{v_{1} + v_{2} + v_{3} + ...... v_{n}}{n}; v_{av} =\tt \sqrt{\frac{8RT}{\pi M}} = \sqrt{\frac{8kT}{\pi m}}

5. Most probable velocity (∝_mp) : It is the velocity possessed by maximum number of molecules of a gas at a given temperature.
\tt \alpha_{mp} = \sqrt{\frac{2RT}{M}} = \sqrt{\frac{2PV}{M}} = \sqrt{\frac{2P}{d}}