Question

At 27 °C, rms velocity of SO$_2$ is x ms$^{-1}$ and most probable velocity of O$_2$ at 127 °C is y ms$^{-1}$. The value of x : y is

• A. 3 : 4
• B. 4 : 3
• C. 2 : 3
• D. 3 : 2

Hint:

Always convert temperatures to Kelvin when using gas laws and kinetic theory formulas. Remember the distinct formulas for different molecular velocities (RMS, most probable, average) and ensure correct molar masses are used. Calculations involving ratios often allow for cancellation of constants like R.

Answer 1

The Correct Option is A

Step 1: Recall the Formulas for RMS Velocity and Most Probable Velocity
The root mean square (rms) velocity ($v_{rms}$) is given by: \[ v_{rms} = \sqrt{\frac{3RT}{M}} \] The most probable velocity ($v_{mp}$) is given by: \[ v_{mp} = \sqrt{\frac{2RT}{M}} \] Where:
$R$ = Ideal gas constant
$T$ = Absolute temperature in Kelvin
$M$ = Molar mass of the gas in kg/mol
Step 2: Convert Temperatures to Kelvin and Calculate Molar Masses
For SO$_2$: Temperature $T_{SO_2} = 27 \text{ °C} = 27 + 273 = 300 \text{ K}$ Molar mass of SO$_2$ ($M_{SO_2}$): S = 32 g/mol, O = 16 g/mol. $M_{SO_2} = 32 + (2 \times 16) = 64 \text{ g/mol} = 64 \times 10^{-3} \text{ kg/mol}$ For O$_2$: Temperature $T_{O_2} = 127 \text{ °C} = 127 + 273 = 400 \text{ K}$ Molar mass of O$_2$ ($M_{O_2}$): O = 16 g/mol. $M_{O_2} = 2 \times 16 = 32 \text{ g/mol} = 32 \times 10^{-3} \text{ kg/mol}$ Step 3: Express x and y using the Velocity Formulas
Given $x$ is the rms velocity of SO$_2$: \[ x = v_{rms(SO_2)} = \sqrt{\frac{3R T_{SO_2}}{M_{SO_2}}} = \sqrt{\frac{3R \times 300}{64 \times 10^{-3}}} \] Given $y$ is the most probable velocity of O$_2$: \[ y = v_{mp(O_2)} = \sqrt{\frac{2R T_{O_2}}{M_{O_2}}} = \sqrt{\frac{2R \times 400}{32 \times 10^{-3}}} \] Step 4: Calculate the Ratio x : y
\[ \frac{x}{y} = \frac{\sqrt{\frac{3R \times 300}{64 \times 10^{-3}}}}{\sqrt{\frac{2R \times 400}{32 \times 10^{-3}}}} \] Combine the square roots and cancel common terms ($R$ and $10^{-3}$): \[ \frac{x}{y} = \sqrt{\frac{3 \times 300}{64} \times \frac{32}{2 \times 400}} \] Simplify the terms inside the square root: \[ \frac{x}{y} = \sqrt{\frac{900}{64} \times \frac{32}{800}} \] Notice that $32$ is half of $64$: \[ \frac{x}{y} = \sqrt{\frac{900}{2 \times 800}} \] \[ \frac{x}{y} = \sqrt{\frac{900}{1600}} \] Simplify the fraction: \[ \frac{x}{y} = \sqrt{\frac{9}{16}} \] \[ \frac{x}{y} = \frac{3}{4} \] Therefore, the ratio $x : y$ is $3 : 4$. Step 5: Analyze the Options
\begin{itemize} \item Option (1): 3 : 4. This matches our calculated ratio. \item Option (2): 4 : 3. Incorrect. \item Option (3): 2 : 3. Incorrect. \item Option (4): 3 : 2. Incorrect. \end{itemize}