At room temperature the rms speed of the molecules of a certain diatomic gas is found to be 1920 m/s. The gas is

Options

A. `H_2`
B. `F_2`
C. `CI_2`
D. `O_2`

Answer

To solve the problem, we need to find the molecular weight of a certain diatomic gas given its root mean square (RMS) speed at room temperature. The steps are as follows:

### Step 1: Write down the formula for RMS speed
The formula for the root mean square speed (v_rms) of gas molecules is given by:

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

where:
- \( v_{rms} \) is the root mean square speed,
- \( R \) is the universal gas constant (approximately \( 8.314 \, \text{J/(mol·K)} \)),
- \( T \) is the absolute temperature in Kelvin,
- \( M \) is the molar mass of the gas in kg/mol.

### Step 2: Substitute known values into the equation
Given:
- \( v_{rms} = 1920 \, \text{m/s} \)
- \( T = 300 \, \text{K} \)
- \( R = 8.314 \, \text{J/(mol·K)} \)

We can substitute these values into the formula:

\[ 1920 = \sqrt{\frac{3 \times 8.314 \times 300}{M}} \]

### Step 3: Square both sides to eliminate the square root
Squaring both sides gives:

\[ 1920^2 = \frac{3 \times 8.314 \times 300}{M} \]

Calculating \( 1920^2 \):

\[ 1920^2 = 3686400 \]

So we have:

\[ 3686400 = \frac{3 \times 8.314 \times 300}{M} \]

### Step 4: Calculate the right-hand side
Now, calculate \( 3 \times 8.314 \times 300 \):

\[ 3 \times 8.314 \times 300 = 7482.6 \]

### Step 5: Rearranging the equation to solve for M
Now we can rearrange the equation to solve for \( M \):

\[ M = \frac{7482.6}{3686400} \]

### Step 6: Calculate M
Calculating \( M \):

\[ M = \frac{7482.6}{3686400} \approx 0.0020298 \, \text{kg/mol} \]

### Step 7: Convert M to grams
To convert \( M \) from kg/mol to g/mol, we multiply by 1000:

\[ M \approx 0.0020298 \times 1000 \approx 2.03 \, \text{g/mol} \]

### Step 8: Identify the gas
The molecular weight of approximately \( 2.03 \, \text{g/mol} \) corresponds to hydrogen (H₂), which has a molecular weight of about \( 2 \, \text{g/mol} \).

### Conclusion
Thus, the gas is **hydrogen**.

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