Question

At what temperature the most probable speed of ${\mathrm{SO}}_2$ molecules have the same value of root mean square speed of ${\mathrm{O}}_2$ molecules?

Answer 1

Step 1: Formula for most probable velocity and root mean square speed

• The formula of most probable velocity of ${\mathrm{SO}}_2$ =$\sqrt{\frac{2\mathrm{RT}}{{\mathrm{M}}_{{\mathrm{SO}}_2}}}$

$$\begin{gathered} \mathrm{Where} \\ \mathrm{R}-\mathrm{gas}\mathrm{constant} \\ \mathrm{T}-\mathrm{Temperature} \\ {\mathrm{M}}_{{\mathrm{SO}}_2}-\mathrm{Molecular}\mathrm{weight}\mathrm{of}{\mathrm{SO}}_2 \end{gathered}$$

• The root mean square speed of${\mathrm{O}}_2=\sqrt{\frac{3\mathrm{RT}}{{\mathrm{M}}_{{\mathrm{O}}_2}}}$

$$\begin{gathered} \mathrm{Where} \\ \mathrm{R}-\mathrm{gas}\mathrm{constant} \\ \mathrm{T}-\mathrm{Temperature} \\ {\mathrm{M}}_{{\mathrm{O}}_2}-\mathrm{Molecular}\mathrm{weight}\mathrm{of}{\mathrm{O}}_2 \end{gathered}$$

Step 2: Equating most probable velocity and root mean square speed

• The most probable speed of ${\mathrm{SO}}_2$ = the root mean square speed of ${\mathrm{O}}_2$

$\sqrt{\frac{2\mathrm{RT}}{{\mathrm{M}}_{{\mathrm{SO}}_2}}}$$=\sqrt{\frac{3\mathrm{RT}}{{\mathrm{M}}_{{\mathrm{O}}_2}}}$

• Squaring on both sides we get,

$\frac{2\mathrm{x}\mathrm{R}\mathrm{x}\mathrm{T}}{{\mathrm{M}}_{{\mathrm{SO}}_2}}=\frac{3\mathrm{x}\mathrm{R}\mathrm{x}\mathrm{T}}{{\mathrm{M}}_{{\mathrm{O}}_2}}$

Step 3: Finding molecular weight:

$$\begin{gathered} \mathrm{The}\mathrm{molecular}\mathrm{weight}\mathrm{of}{\mathrm{SO}}_2=32\mathrm{x}2\left(16\right) \\ =64 \\ \mathrm{The}\mathrm{molecular}\mathrm{weight}\mathrm{of}{\mathrm{O}}_2=2\left(16\right) \\ =32 \end{gathered}$$

Step 4: Substituting values:

$$\begin{gathered} \frac{2\mathrm{x}\mathrm{R}\mathrm{x}\mathrm{T}}{{\mathrm{M}}_{{\mathrm{SO}}_2}}=\frac{3\mathrm{x}\mathrm{R}\mathrm{x}\mathrm{T}}{{\mathrm{M}}_{{\mathrm{O}}_2}} \\ \frac{2\mathrm{x}\mathrm{R}\mathrm{x}\mathrm{T}}{64}=\frac{3\mathrm{x}\mathrm{R}\mathrm{x}300}{32} \\ \frac{2\mathrm{T}}{64}=\frac{900}{32} \\ \mathrm{T}=900\mathrm{K} \end{gathered}$$

Hence, at $900\mathrm{K}$ the most probable speed of ${\mathrm{SO}}_2$ molecules have the same value of root mean square speed of ${\mathrm{O}}_2$ molecules