Question

At which temperature the r.m.s. velocity of a hydrogen molecule equal to that of an oxygen molecule at 47ºC?

• A. 80 K
• B. -73 K
• C. 20 K
• D. 4 K

Answer 1

The Correct Option is C

Using the Formula for Root Mean Square (r.m.s.) Velocity: The r.m.s. velocity \( v_{rms} \) for a gas is given by:

\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]

where \( R \) is the gas constant, \( T \) is the temperature, and \( M \) is the molar mass of the gas.

Set up the Equation for Hydrogen and Oxygen: To find the temperature at which the r.m.s. velocity of hydrogen equals that of oxygen at 47°C, we set:

\[ \sqrt{\frac{3RT_{H_2}}{M_{H_2}}} = \sqrt{\frac{3RT_{O_2}}{M_{O_2}}} \]

Isolate \( T_{H_2} \): Square both sides to remove the square root:

\[ \frac{3RT_{H_2}}{M_{H_2}} = \frac{3RT_{O_2}}{M_{O_2}} \]

Simplify by canceling \( 3R \) on both sides:

\[ T_{H_2} = T_{O_2} \times \frac{M_{H_2}}{M_{O_2}} \]

Substitute Values for Molar Mass and Temperature: Given \( T_{O_2} = 47°C = 320 \, K \),

\[ T_{H_2} = 320 \times \frac{2}{32} = 20 \, K \]

Answer 2

To find the temperature at which the root mean square (r.m.s.) velocity of a hydrogen molecule equals that of an oxygen molecule at 47°C, we use the formula for r.m.s. velocity:

\(v_{\text{rms}} = \sqrt{\frac{3kT}{m}}\)

where:

• \(v_{\text{rms}}\) is the root mean square velocity,
• \(k\) is the Boltzmann constant,
• \(T\) is the temperature in Kelvin,
• \(m\) is the molecular mass.

 

Given that the r.m.s velocity of the hydrogen molecule is equal to that of the oxygen molecule, we can set up the equation:

\(\sqrt{\frac{3kT_{\text{H}_2}}{m_{\text{H}_2}}} = \sqrt{\frac{3kT_{\text{O}_2}}{m_{\text{O}_2}}}\)

From this equation, we cancel out the common terms:

\(\frac{T_{\text{H}_2}}{m_{\text{H}_2}} = \frac{T_{\text{O}_2}}{m_{\text{O}_2}}\)

Rearranging gives us:

\(T_{\text{H}_2} = T_{\text{O}_2} \times \frac{m_{\text{H}_2}}{m_{\text{O}_2}}\)

The molecular masses are:

• Molecular mass of hydrogen, \(m_{\text{H}_2} = 2 \text{ u}\).
• Molecular mass of oxygen, \(m_{\text{O}_2} = 32 \text{ u}\).

 

The temperature of the oxygen molecule, \(T_{\text{O}_2}\), must be in Kelvin. 47°C is converted to Kelvin as follows:

\(T_{\text{O}_2} = 47 + 273 = 320 \text{ K}\)

Substituting these values into the equation:

\(T_{\text{H}_2} = 320 \times \frac{2}{32}\)

Simplifying gives:

\(T_{\text{H}_2} = 320 \times \frac{1}{16} = 20 \text{ K}\)

Therefore, the correct answer is 20 K.