The ratio of root mean square velocity of hydrogen at K 50 to that of nitrogen at K 500 is closest to

Options

A. 1.18
B. 0.85
C. 0.59
D. 1.4

Explanation

To find the ratio of the root mean square (RMS) velocity of hydrogen (H₂) at 50 K to that of nitrogen (N₂) at 500 K, we can use the formula for the root mean square velocity:

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

Where:
- \( v_{rms} \) = root mean square velocity
- \( R \) = universal gas constant
- \( T \) = absolute temperature in Kelvin
- \( M \) = molar mass of the gas in kg/mol

### Step 1: Write the formula for the RMS velocity of hydrogen and nitrogen

For hydrogen (H₂):
\[ v_{rms, H₂} = \sqrt{\frac{3RT_{H₂}}{M_{H₂}}} \]

For nitrogen (N₂):
\[ v_{rms, N₂} = \sqrt{\frac{3RT_{N₂}}{M_{N₂}}} \]

### Step 2: Substitute the values into the equations

Given:
- \( T_{H₂} = 50 \, K \)
- \( T_{N₂} = 500 \, K \)
- Molar mass of hydrogen \( M_{H₂} = 2 \, g/mol = 0.002 \, kg/mol \)
- Molar mass of nitrogen \( M_{N₂} = 28 \, g/mol = 0.028 \, kg/mol \)

Substituting these values:

\[ v_{rms, H₂} = \sqrt{\frac{3R \cdot 50}{0.002}} \]

\[ v_{rms, N₂} = \sqrt{\frac{3R \cdot 500}{0.028}} \]

### Step 3: Calculate the ratio of the RMS velocities

Now, we need to find the ratio \( \frac{v_{rms, H₂}}{v_{rms, N₂}} \):

\[ \frac{v_{rms, H₂}}{v_{rms, N₂}} = \frac{\sqrt{\frac{3R \cdot 50}{0.002}}}{\sqrt{\frac{3R \cdot 500}{0.028}}} \]

### Step 4: Simplify the ratio

This simplifies to:

\[ \frac{v_{rms, H₂}}{v_{rms, N₂}} = \sqrt{\frac{50 \cdot 0.028}{500 \cdot 0.002}} \]

Calculating the values inside the square root:

\[ = \sqrt{\frac{50 \cdot 0.028}{500 \cdot 0.002}} = \sqrt{\frac{1.4}{1}} = \sqrt{1.4} \]

### Step 5: Calculate the square root

Calculating \( \sqrt{1.4} \):

\[ \sqrt{1.4} \approx 1.183 \]

### Final Answer

Thus, the ratio of the root mean square velocity of hydrogen at 50 K to that of nitrogen at 500 K is approximately **1.18**.

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