# Melting and Boiling Points, Molecular Speeds

Most Probable Velocity Cp (or) Vp:
\tt C_p=\sqrt{\frac{2RT}{M}}=\sqrt{\frac{2PV}{M}}=\sqrt{\frac{2P}{d}}

RMS Velocity (C or u):
\tt u=\sqrt{\frac{u_1^2+u_2^2+u_3^2+...+u_n^2}{n}}
\tt PV=\frac{1}{3}\ MC^2
\tt C=\sqrt{\frac{3PV}{M}}=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3P}{d}}

\tt C=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3\times 8.314\times 10^{-7}\times T}{M}}
\tt C=1.58\sqrt{\frac{T}{M}}\times 10^4 cm/sec

Relation between Cp : \tt \overline{C} : C
\tt =\sqrt{\frac{2RT}{M}}:\sqrt{\frac{8RT}{M}}:\sqrt{{\frac{3RT}{M}}}
\tt =\sqrt{2}:\sqrt{\frac{8}{\pi}}:\sqrt{3}
= 1.414 : 1.596 : 1.732

Cp = 0.816 × C
\tt \overline{C} = 0.9213 × C
\tt \frac{C_1}{C_2}=\sqrt{\frac{T_1}{T_2}\cdot\frac{M_2}{M_1}}

### Part2: View the Topic in this Video from 0:02 to 7:21

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1. Root mean square velocity (urms) : 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 (vav) : 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}}