Question

The rms speeds of the molecules of Hydrogen, Oxygen and Carbondioxide at the same temperature are $V_H$, $V_O$ and $V_C$ respectively then :

• A. $V_H = V_O = V_C$
• B. $V_H = V_O>V_C$
• C. $V_H>V_O>V_C$
• D. $V_C>V_O>V_H$

Hint:

At the same temperature, lighter molecules move faster. Hydrogen being the lightest gas will always have the highest rms speed compared to oxygen or carbon dioxide.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The root mean square (rms) speed of a gas molecule is determined by the kinetic theory of gases, which states that at a fixed temperature, the speed is inversely proportional to the square root of the molar mass.
Step 2: Key Formula or Approach:
The formula for rms speed is:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where \(R\) is the universal gas constant, \(T\) is the absolute temperature, and \(M\) is the molar mass of the gas.
Step 3: Detailed Explanation:
Since all gases are at the same temperature \(T\), we can observe the relationship:
\[ v_{rms} \propto \frac{1}{\sqrt{M}} \]
Now, let us compare the molar masses of the given gases:
1. Hydrogen (\(H_2\)): \(M_H = 2 \text{ g/mol}\)
2. Oxygen (\(O_2\)): \(M_O = 32 \text{ g/mol}\)
3. Carbon dioxide (\(CO_2\)): \(M_C = 44 \text{ g/mol}\)
Since \(M_H<M_O<M_C\), the inverse relationship for speeds will be:
\[ V_H>V_O>V_C \]
Step 4: Final Answer:
The correct relationship between the rms speeds is $V_H>V_O>V_C$.