At room temperature, the rms speed of the molecules of a certain diatomic gas is found to be `1930 m//s`. The gas is

Options

A. `H_(2)`
B. `F_(2)`
C. `CL_(2)`
D. CL_(2)`

Answer

To determine the diatomic gas based on the given root mean square (rms) speed of its molecules, we can use the formula for rms speed:

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

Where:
- \( v_{rms} \) = rms speed of the gas molecules
- \( R \) = universal gas constant (approximately \( 8.31 \, \text{J/(mol K)} \))
- \( T \) = absolute temperature in Kelvin
- \( M \) = molar mass of the gas in kg/mol

### Step 1: Rearranging the formula to find M
To find the molar mass \( M \), we can rearrange the formula:

\[ M = \frac{3RT}{v_{rms}^2} \]

### Step 2: Substituting the known values
Given:
- \( v_{rms} = 1930 \, \text{m/s} \)
- \( R = 8.31 \, \text{J/(mol K)} \)
- \( T = 300 \, \text{K} \)

Now, substituting these values into the equation:

\[ M = \frac{3 \times 8.31 \times 300}{(1930)^2} \]

### Step 3: Calculating the numerator
Calculating the numerator:

\[ 3 \times 8.31 \times 300 = 7479 \]

### Step 4: Calculating the denominator
Calculating the denominator:

\[ (1930)^2 = 3724900 \]

### Step 5: Completing the calculation for M
Now substituting the values into the equation for \( M \):

\[ M = \frac{7479}{3724900} \approx 0.00201 \, \text{kg/mol} \]

### Step 6: Converting kg/mol to g/mol
To convert \( M \) from kg/mol to g/mol, we multiply by 1000:

\[ M \approx 0.00201 \times 1000 \approx 2.01 \, \text{g/mol} \]

### Step 7: Identifying the gas
The molar mass of approximately \( 2.01 \, \text{g/mol} \) corresponds to hydrogen gas \( H_2 \), which has a molar mass of about \( 2 \, \text{g/mol} \).

### Conclusion
Thus, the gas is \( H_2 \) (hydrogen gas).

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