MHT CET · Physics · Kinetic Theory of Gases
The velocity of 4 gas molecules are given by \(1 \mathrm{~km} / \mathrm{s}, 3 \mathrm{~km} / \mathrm{s}, 5 \mathrm{~km} / \mathrm{s}\) and \(7 \mathrm{~km} / \mathrm{s}\). Calculate the difference between average and RMS velocity.
- A \(0.338\)
- B \(0.438\)
- C \(0.583\)
- D \(0.683\)
Answer & Solution
Correct Answer
(C) \(0.583\)
Step-by-step Solution
Detailed explanation
The average velocity
\(
\begin{aligned}
v_{\mathrm{av}} &=\frac{v_{1}+v_{2}+v_{3} \ldots v_{n}}{N} \\
&=\frac{1+3+5+7}{4}=4 \mathrm{~km} / \mathrm{s}
\end{aligned}
\)
Root mean square velocity
\(
v_{\mathrm{rms}}=\sqrt{\frac{v_{1}^{2}+v_{2}^{2}+v_{3}^{2}+v_{4}^{2}+\ldots+v_{n}^{2}}{N}}
\)
\(
\begin{array}{l}
=\sqrt{\frac{1+(3)^{2}+(5)^{2}+(7)^{2}}{4}} \\
=\sqrt{21}=4.583 \mathrm{~km} / \mathrm{s}
\end{array}
\)
Difference between average velocity and root mean square velocity \(=4.583-4\)
\(
=0.583 \mathrm{~km} / \mathrm{s}
\)
\(
\begin{aligned}
v_{\mathrm{av}} &=\frac{v_{1}+v_{2}+v_{3} \ldots v_{n}}{N} \\
&=\frac{1+3+5+7}{4}=4 \mathrm{~km} / \mathrm{s}
\end{aligned}
\)
Root mean square velocity
\(
v_{\mathrm{rms}}=\sqrt{\frac{v_{1}^{2}+v_{2}^{2}+v_{3}^{2}+v_{4}^{2}+\ldots+v_{n}^{2}}{N}}
\)
\(
\begin{array}{l}
=\sqrt{\frac{1+(3)^{2}+(5)^{2}+(7)^{2}}{4}} \\
=\sqrt{21}=4.583 \mathrm{~km} / \mathrm{s}
\end{array}
\)
Difference between average velocity and root mean square velocity \(=4.583-4\)
\(
=0.583 \mathrm{~km} / \mathrm{s}
\)
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