Question

The velocity of three molecules are 3V, 4V and 5V respectively. Their rms speed will be:
A. $\dfrac{3}{{15}}V$
B. $\sqrt {\dfrac{3}{{50}}} V$
C. $\dfrac{{50}}{3}V$
D. $\sqrt {\dfrac{{50}}{3}} V$

Answer 1

Hint:The square root of the average velocity-squared of the molecules in a gas is used to calculate the root-mean-square speed of particles in a gas.The reason we use the rms velocity rather than the average is that the net velocity of a typical gas sample is zero because the particles are travelling in all directions.

Complete step by step answer:
We know that;
${V_{rms}}$ (root mean square velocity) $ = \sqrt {\dfrac{{v_1^2 + v_2^2 + ...... + v_n^2}}{n}} $ (standard result)
Now, we will write all the given values in the above given question accordingly; Let us consider that the velocity of the three molecules are given as ${v_1},{v_2}$ and ${v_3}$ respectively.Hence we can write it as;
${v_1} = 3V$
$\Rightarrow {v_2} = 4V$
$\Rightarrow {v_3} = 5V$
As a result, we may acquire the necessary rms speed by plugging all three numbers into the above-mentioned calculation and equating.
${V_{rms}} = \sqrt {\dfrac{{{3^2} + {4^2} + {5^2}}}{3}} \\
\therefore {V_{rms}} = \sqrt {\dfrac{{50}}{3}} V $
Therefore, their rms speed will be $\sqrt {\dfrac{{50}}{3}} V$

So, the correct option is D.

Note:The very important significance of rms velocity is because we can't find the velocity of each individual molecule in the gas (because all the individual point masses, i.e. molecules, are travelling at all conceivable speeds in all possible directions), we assume that all molecules are flowing at a constant velocity. It's important to remember that the gas's total energy must remain constant.