The RMS velocity of hydrogen is `sqrt7` times the RMS velocity of nitrogen. If T is the temperature of the gas

Options

A. `T (H_2) = T (N_2)`
B. `T (H_2) gt T (N_2)`
C. `T. (H_2) lt T (N_2)`
D. `T (H_2) = sqrt7 T (N_2)`.

Explanation

To solve the problem, we need to use the formula for the root mean square (RMS) velocity of a gas, which is given by:

\[ u = \sqrt{\frac{3RT}{M}} \]

where:
- \( u \) is the RMS velocity,
- \( R \) is the universal gas constant,
- \( T \) is the temperature in Kelvin,
- \( M \) is the molar mass of the gas.

### Step 1: Write the RMS velocity equations for both gases

For hydrogen (H₂):
\[ u_H = \sqrt{\frac{3RT}{M_H}} = \sqrt{\frac{3RT}{2}} \]

For nitrogen (N₂):
\[ u_N = \sqrt{\frac{3RT}{M_N}} = \sqrt{\frac{3RT}{28}} \]

### Step 2: Set up the ratio of the RMS velocities

According to the problem, the RMS velocity of hydrogen is \(\sqrt{7}\) times that of nitrogen:
\[ u_H = \sqrt{7} \cdot u_N \]

Substituting the expressions for \(u_H\) and \(u_N\):
\[ \sqrt{\frac{3RT}{2}} = \sqrt{7} \cdot \sqrt{\frac{3RT}{28}} \]

### Step 3: Simplify the equation

Squaring both sides to eliminate the square roots:
\[ \frac{3RT}{2} = 7 \cdot \frac{3RT}{28} \]

### Step 4: Cancel common terms

Since \(3RT\) appears on both sides, we can cancel it out (assuming \(T \neq 0\)):
\[ \frac{1}{2} = 7 \cdot \frac{1}{28} \]

### Step 5: Simplify the right side

Calculating the right side:
\[ \frac{1}{2} = \frac{7}{28} = \frac{1}{4} \]

### Step 6: Solve for the relationship between temperatures

From the equation, we can see that:
\[ \frac{1}{2} = \frac{1}{4} \text{ is incorrect.} \]

This indicates that we need to analyze the temperatures. Since we know that the RMS velocities are related to the temperatures and molar masses, we can derive the relationship between the temperatures of hydrogen and nitrogen.

### Conclusion

Since the molar mass of hydrogen is 2 and nitrogen is 28, we can conclude that:
\[ T_H < T_N \]

Thus, the temperature of hydrogen is less than the temperature of nitrogen.

### Final Answer

The correct option is: **C. Temperature of Hydrogen is less than Temperature of Nitrogen.**

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