A sample of \(\mathrm{H}_{2} \mathrm{SO}_{4}\) contains 2.02 \(\mathrm{g}\) of hydrogen, 32.07 \(\mathrm{g}\) of sulfur, and 64.00 \(\mathrm{g}\) of oxygen. How many grams of sulfur and grams of oxygen are present in a second sample of \(\mathrm{H}_{2} \mathrm{SO}_{4}\) containing 7.27 \(\mathrm{g}\) of hydrogen?

To find these ratios, we need to divide the grams of each element in the sample by the grams of hydrogen. So, the ratio of hydrogen to sulfur will be: \[ \frac{\text{grams of hydrogen}}{\text{grams of sulfur}} = \frac{2.02}{32.07} \] Similarly, the ratio of hydrogen to oxygen will be: \[ \frac{\text{grams of hydrogen}}{\text{grams of oxygen}} = \frac{2.02}{64.00} \]

Now, let's calculate the ratios for the first sample: \[ \frac{\text{grams of hydrogen}}{\text{grams of sulfur}} = \frac{2.02}{32.07} \approx 0.063 \] \[ \frac{\text{grams of hydrogen}}{\text{grams of oxygen}} = \frac{2.02}{64.00} \approx 0.0316 \]

We can use these ratios to find the grams of sulfur and oxygen in the second sample, which contains 7.27 grams of hydrogen. To find the grams of sulfur in the second sample, multiply the ratio we found for hydrogen to sulfur by the grams of hydrogen in the second sample: \[ \text{grams of sulfur} = \text{grams of hydrogen} \times \frac{\text{grams of sulfur}}{\text{grams of hydrogen}} = 7.27 \times 0.063 \approx 0.458 \] To find the grams of oxygen in the second sample, multiply the ratio we found for hydrogen to oxygen by the grams of hydrogen in the second sample: \[ \text{grams of oxygen} = \text{grams of hydrogen} \times \frac{\text{grams of hydrogen}}{\text{grams of oxygen}} = 7.27 \times 0.0316 \approx 0.230 \]

Based on the ratios calculated in the first sample, the second sample containing 7.27 grams of hydrogen has approximately 0.458 grams of sulfur and 0.230 grams of oxygen.