Question

The r.m.s speed of hydrogen gas molecules at a certain temperature is $200m{{s}^{-1}}$ . If the absolute temperature of the gas is doubled and the hydrogen gas dissociates into atomic hydrogen, then the r.m.s. speed will become:
A. $200m{{s}^{-1}}$
B. $400m{{s}^{-1}}$
C. $600m{{s}^{-1}}$
D. $800m{{s}^{-1}}$

Answer 1

Hint: The r.m.s speed of a gas depends directly on the square root of the absolute temperature of the gas and inversely on the square root of the molar mass of the molecules of the gas. This problem can be solved by finding out the molar mass of molecules and absolute temperature in the final condition and comparing with the initial condition to get the final r.m.s speed of the gas in terms of the initial speed.
Formula used:
${{v}_{rms}}=\sqrt{\dfrac{3RT}{M}}$
where R is the universal gas constant equal to $8.314J.mo{{l}^{-1}}.{{K}^{-1}}$, T is the absolute temperature of the gas and M is the molar mass of the molecules of the gas.

Complete Step-by-Step solution:
The molecules of a gas have a certain speed. There are in fact, three kinds of speeds of gas molecules: The average speed, most probable speed and root mean square (rms) speed. Of these the rms speed is the most significant in defining the energy and properties of the gas.
The rms speed is the root of the mean of the squares of all the molecules in the gas. It is determined by the absolute temperature and the molar mass of the gas. Its mathematical expression is given by
${{v}_{rms}}=\sqrt{\dfrac{3RT}{M}}$ --(1)
where R is the universal gas constant equal to $8.314J.mo{{l}^{-1}}.{{K}^{-1}}$, T is the absolute temperature of the gas and M is the molar mass of the molecules of the gas.
This problem can be solved by finding out the molar mass of molecules and absolute temperature in the final condition and comparing with the initial condition to get the final r.m.s speed of the gas in terms of the initial speed.
Let us analyse the information given to us.
r.m.s speed of molecular hydrogen $\left( {{v}_{1}} \right)=200m{{s}^{-1}}$ --(2)
Let the absolute temperature initially be T. --(3)
Let the molar mass of molecular hydrogen be M. --(4)
Now, let the
r.m.s speed in the final condition (atomic hydrogen) be ${{v}_{2}}$. --(5)
The absolute temperature is doubled from initial condition. It is $2T$. --(6)
Now, hydrogen is a diatomic gas and dissociates into two atoms of hydrogen.
Therefore, the molar mass of hydrogen atom should be half of molecular hydrogen. Thus, it is $\dfrac{M}{2}$. --(7)
Using (1),(2),(3),(4),(5),(6) and (7),
$\dfrac{{{v}_{2}}}{{{v}_{1}}}=\dfrac{\sqrt{\dfrac{3R2T}{\dfrac{M}{2}}}}{\sqrt{\dfrac{3RT}{M}}}$
$\therefore \dfrac{{{v}_{2}}}{200}=\dfrac{\sqrt{\dfrac{4\times 3RT}{M}}}{\sqrt{\dfrac{3RT}{M}}}$
$\therefore \dfrac{{{v}_{2}}}{200}=\sqrt{4}=2$
$\therefore {{v}_{2}}=2\times 200=400m{{s}^{-1}}$.
Hence, the required r.m.s speed is $400m{{s}^{-1}}$.
So, the correct option is B) $400m{{s}^{-1}}$.

Note: One should always be careful in problems especially pertaining to thermodynamics that the temperature value taken should always be the absolute temperature that is the Kelvin scale value of the temperature. Taking other scales such as the temperature in Celsius or Fahrenheit will give erroneous results and lead to a lot of confusion for the student. Sometimes questions are purposefully set with the temperatures given in Celsius to fool unmindful students and to lead them into committing a silly mistake. So, one should always read the question carefully to determine whether the temperature given is in Kelvin or not. If not, it should always be converted into Kelvin first before proceeding, even if the answer at the end required is in Celsius. The final answer can be obtained by converting from Kelvin to Celsius in such situations.