Question

Find the r.m.s speed of hydrogen molecule at room temperature (300K)

Answer 1

Hint: The root mean square speed is the measure of the speed of particles in gas, defined as the square root of average velocity squared of the molecules in gas. The root mean square speed takes into account both temperature and molecular mass which directly affects the kinetic energy of gas.

Complete answer:
Now, from the question we have,
The root mean square speed is given by,
${V_{rms}} = \dfrac{{\sqrt {3RT} }}{M}$, where
R- Rydbergs’ constant
T- temperature
M- mass
Given,
T = 300K
R = 8.3
M = 2gm
Mass of 1 mole of hydrogen gas = 2gm = $2 \times {10^{ - 3}}$kg
${V_{rms}} = \dfrac{{\sqrt {3RT} }}{M} = \dfrac{{\sqrt {3 \times 8.3 \times 300} }}{{2 \times {{10}^{ - 3}}}}$
${V_{rms}} = 1.93 \times {10^3}\dfrac{m}{s}$
Therefore the rms speed of hydrogen molecule at room temperature is ${V_{rms}} = 1.93 \times {10^3}\dfrac{m}{s}$

Note:
The temperature of a body is the measure of the average kinetic energy of a body. We should note that the temperature of a body always depends upon its average kinetic energy and since the average kinetic energy can have a minimum possible value of zero, therefore an object cannot be cooled below a certain minimum value, this value is known as absolute zero. This is the lowest possible temperature in our universe and no object could be cooled to this temperature, it is equivalent to -273 degree Celsius or 0 Kelvin. These two scales of temperature can be converted with the following expression,

Temperature in Kelvin = Temperature in degree Celsius + 273

As absolute zero is equal to 0 in the Kelvin scale, it is also known as the absolute scale. In a gas the particles are always in a state of random motion, all the particles move at different speeds constantly colliding and changing their speed and direction.

As the particles collide and change velocity it is not practical to measure each velocity, and as there are particles moving in one direction equal to the particles moving in the opposite direction they cancel out and the average velocity will be equal to zero, hence there is an alternate way to determine their average velocity. That way is root mean square speed.