Understanding Molecular Speeds in Gases

For a large number of gas molecules at a given temperature, there are three key types of speeds: the most probable speed (C*), the average speed ($\overline{\mathrm{C}}$), and the root mean square speed (C, which is the square root of the mean square speed). These speeds are derived from the Maxwell-Boltzmann distribution of molecular speeds and have fixed mathematical relationships.

Step-by-Step Solution:

Step 1: Recall the Formulas

The formulas for these speeds in terms of the gas constant (R), molar mass (M), and temperature (T) are:

• Most Probable Speed, C* = $\sqrt{\frac{2RT}{M}}$
• Average Speed, $\overline{\mathrm{C}}$ = $\sqrt{\frac{8RT}{\pi M}}$
• Root Mean Square Speed, C = $\sqrt{\frac{3RT}{M}}$

Step 2: Find the Ratios

To find the ratios C* : $\overline{\mathrm{C}}$ : C, we can divide each speed by the value of C* to simplify. Notice that the common factor $\sqrt{\frac{RT}{M}}$ is present in all three formulas.

• Let C* = $\sqrt{2}$
• Then $\overline{\mathrm{C}}$ = $\sqrt{\frac{8}{\pi }}$ ≈ $\sqrt{2.546}$ ≈ $1.595$
• And C = $\sqrt{3}$ ≈ $1.732$

Now, we express these values as ratios relative to C* (which is 1).

• C* : $\overline{\mathrm{C}}$ : C = 1 : $\frac{\overline{\mathrm{C}}}{\text{C*}}$ : $\frac{\text{C}}{\text{C*}}$
• 1 : $\frac{\sqrt{\frac{8}{\pi }}}{\sqrt{2}}$ : $\frac{\sqrt{3}}{\sqrt{2}}$

Step 3: Simplify the Ratios

Let's calculate the numerical values of these ratios:

• $\frac{\overline{\mathrm{C}}}{\text{C*}}$ = $\frac{\sqrt{\frac{8}{\pi }}}{\sqrt{2}}$ = $\sqrt{\frac{8}{2\pi }}$ = $\sqrt{\frac{4}{\pi }}$ = $\frac{2}{\sqrt{\pi }}$ ≈ $\frac{2}{1.772}$ ≈ $1.128$
• $\frac{\text{C}}{\text{C*}}$ = $\frac{\sqrt{3}}{\sqrt{2}}$ = $\sqrt{\frac{3}{2}}$ ≈ $\sqrt{1.5}$ ≈ $1.225$

Therefore, the ratio is C* : $\overline{\mathrm{C}}$ : C = 1 : 1.128 : 1.225

Final Answer: The correct option is C* : $\overline{\mathrm{C}}$ : C = 1 : 1.128 : 1.225

Related Topics & Formulae:

Key Concepts:

• Maxwell-Boltzmann Distribution: This is the probability distribution that describes the speeds of particles in a gas. It shows that while molecules have a wide range of speeds, most are clustered around the most probable speed.
• Kinetic Theory of Gases: This theory explains the macroscopic properties of gases (like pressure and temperature) by considering their molecular composition and motion. These speed definitions are central to this theory.

Important Formulae:

• Most Probable Speed: $C_{\text{*}}=\sqrt{\frac{2RT}{M}}$
• Average Speed: $\overline{C}=\sqrt{\frac{8RT}{\pi M}}$
• Root Mean Square Speed: $C_{\text{rms}}=\sqrt{\frac{3RT}{M}}$

Note: The root mean square speed (C) is directly related to the average kinetic energy of the gas molecules: $\frac{1}{2}m{C_{\text{rms}}}^2=\frac{3}{2}kT$, where $m$ is the mass of one molecule and $k$ is the Boltzmann constant.