At room temperature a diatomic gas is found to have an r.m.s. speed of 1930 ms$$^{-1}$$. The gas is:

The root mean square (r.m.s.) speed of a gas is given by the formula:

$$v_{rms} = \sqrt{\frac{3RT}{M}}$$

where $$ R $$ is the universal gas constant, $$ T $$ is the temperature in Kelvin, and $$ M $$ is the molar mass of the gas in kg/mol. The r.m.s. speed is provided as 1930 m/s, and room temperature is taken as 300 K. The gas constant $$ R $$ is 8.314 J/(mol·K), which is equivalent to 8.314 kg·m²/(s²·mol·K) since 1 J = 1 kg·m²/s².

To identify the gas, we rearrange the formula to solve for the molar mass $$ M $$:

$$v_{rms}^2 = \frac{3RT}{M} \implies M = \frac{3RT}{v_{rms}^2}$$

Substituting the given values:

$$M = \frac{3 \times 8.314 \times 300}{(1930)^2}$$

First, compute the numerator:

$$3 \times 8.314 = 24.942$$ $$24.942 \times 300 = 7482.6$$

So the numerator is 7482.6 kg·m²/(s²·mol).

Next, compute the denominator, which is $$ v_{rms}^2 $$:

$$1930^2 = 1930 \times 1930$$

Calculating this:

$$1930 \times 1930 = (2000 - 70)^2 = 2000^2 - 2 \times 2000 \times 70 + 70^2 = 4,000,000 - 280,000 + 4,900 = 3,724,900$$

So $$ v_{rms}^2 = 3,724,900 $$ m²/s².

Now, substitute back into the formula for $$ M $$:

$$M = \frac{7482.6}{3,724,900} \text{ kg/mol}$$

Performing the division:

$$M = \frac{7482.6}{3724900} \approx 0.0020088 \text{ kg/mol}$$

To convert to grams per mole (g/mol), multiply by 1000:

$$M = 0.0020088 \times 1000 = 2.0088 \text{ g/mol}$$

The molar mass is approximately 2.009 g/mol. Comparing this with the molar masses of the given diatomic gases:

• Hydrogen (H₂): 2 g/mol
• Chlorine (Cl₂): 71 g/mol (since Cl is 35.5 g/mol)
• Oxygen (O₂): 32 g/mol
• Fluorine (F₂): 38 g/mol

The calculated molar mass of 2.009 g/mol is very close to the molar mass of hydrogen gas (H₂), which is 2 g/mol. The slight difference can be attributed to approximations in room temperature or the gas constant, but it is evident that hydrogen is the gas.

To verify, the r.m.s. speed for hydrogen at 300 K can be calculated:

$$v_{rms} = \sqrt{\frac{3 \times 8.314 \times 300}{0.002}} = \sqrt{\frac{7482.6}{0.002}} = \sqrt{3,741,300} \approx 1934.5 \text{ m/s}$$

This is close to the given 1930 m/s, considering typical approximations. The other gases have much lower r.m.s. speeds at the same temperature, confirming that hydrogen is the correct gas.

Hence, the correct answer is Option A.