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The RMS speed of hydrogen molecules at room temperature is $2400m/s$. At room temperature what will be the RMS speed of the oxygen molecules?A.$400m/s$B.$300m/s$C.$600m/s$D.$1600m/s$

Last updated date: 13th Jun 2024
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Hint:
According to the kinetic theory of matter, the molecules of a gas are always in a state of ceaseless motion with a frequent collision with each other. As the molecules collide there is a transfer of energy from one molecule to the other. Due to this transfer of energy, the molecules which gain energy their velocity increases, while which loses energy their velocity decreases.

Step 1
As we study the different properties of gases, there are three types of molecular velocities about which we need to know.
They are:
Average velocity – At a particular temperature average velocity is the arithmetic mean of the different velocities of different molecules of the gas. From Maxwell’s equation, the mathematical expression for average velocity is given by $\mathop u\limits^ - = \sqrt {\frac{{8RT}}{{\pi M}}}$where M is the molecular mass of the gas, R is the Universal gas constant and T is the absolute temperature.
Root mean-square velocity – At a particular temperature the square root of the arithmetic mean of the squares of velocities for different molecules of a gas is known as the root mean square velocity or RMS velocity of the gas. The value of RMS velocity is calculated with the help of a kinetic gas equation and is given by $u = \sqrt {\frac{{3RT}}{M}}$.
Most probable velocity – at a particular temperature the velocity possessed by the majority fraction of the molecules is known as the most probable velocity. The mathematical expression for the most probable velocity is given by $\alpha = \sqrt {\frac{{2RT}}{M}}$.
Step 2
In the given question the rms velocity of hydrogen gas is $2400m/s$ and the molecular mass for hydrogen is $2$.
We know, $u = \sqrt {\frac{{3RT}}{M}}$
$\therefore \sqrt {\frac{{3RT}}{2}} = 2400$
Taking square on both sides, we have
$\therefore \frac{{3RT}}{2} = 2400 \times 2400$
$\therefore RT = 2400 \times 2400 \times \frac{2}{3}$
Step 3
For oxygen molecules the molecular mass is $32$.
$u = \sqrt {\frac{{3RT}}{M}}$
Putting the value of RT we get,
$\therefore u = \sqrt {\frac{{3 \times 2400 \times 2400 \times 2}}{{32 \times 3}}}$
$\therefore u = 6 \times 100$
$\therefore u = 600m/s$

Hence, we can find that it is option c) which is the correct answer for the given question.

Note:With the increase in temperature, more molecules have higher velocities and fewer will have lower velocities and vice-versa. At absolute zero, the molecules of gas become perfectly motionless and become nonexistent. Thus the absolute scale of temperature is always positive.