- Identify given values: \(V_1 = 230\text{ mL}\), \(V_2 = 860\text{ mL}\), \(n_2 = 3.8 \times 10^{-4}\text{ mol}\).
- Use the proportion \(\frac{V_1}{n_1} = \frac{V_2}{n_2}\) to find \(n_1\).
- Rearrange to solve for \(n_1 = \frac{V_1 \cdot n_2}{V_2}\).
- Substitute and calculate: \(n_1 \approx 1.0 \times 10^{-4}\text{ mol}\).
- Final Answer: \(\boxed{1.0 \times 10^{-4}\text{ mol}}\).
Explanation
- Problem Analysis and Data Identification
To determine the initial quantity of helium in the balloon, we identify the given data and the relationship between the variables involved. We are given the initial volume \(V_1 = 230\text{ mL}\) and the final volume \(V_2 = 860\text{ mL}\). The amount of gas in the expanded balloon is given as \(n_2 = 3.8 \times 10^{-4}\text{ mol}\). Since the temperature and pressure are constant, the volume of a gas is directly proportional to the amount of gas (moles) present, which is described by the relation:
\[\frac{V_1}{n_1} = \frac{V_2}{n_2}\]
where \(n_1\) is the initial quantity of helium in moles that we need to find.
- Algebraic Rearrangement and Substitution
We rearrange the given equation \(\frac{V_1}{n_1} = \frac{V_2}{n_2}\) to isolate the unknown variable \(n_1\). By cross-multiplying, we get:
\[V_1 \cdot n_2 = V_2 \cdot n_1\]
Dividing both sides by \(V_2\), we obtain the formula for the initial moles:
\[n_1 = \frac{V_1 \cdot n_2}{V_2}\]
Now, we substitute the known values into this equation:
\[n_1 = \frac{230\text{ mL} \times (3.8 \times 10^{-4}\text{ mol})}{860\text{ mL}}\]
- Final Calculation and Conclusion
Using the substituted values, we calculate the initial moles of helium:
\[n_1 = \frac{874 \times 10^{-4}\text{ mol}}{860} \approx 1.016 \times 10^{-4}\text{ mol}\]
Comparing this result to the given options, we find that \(1.016 \times 10^{-4}\text{ mol}\) is closest to \(1.0 \times 10^{-4}\text{ mol}\). Thus, the initial quantity of helium present in the balloon was approximately \(1.0 \times 10^{-4}\text{ mol}\). The final answer is \(\boxed{1.0 \times 10^{-4}\text{ mol}}\).
Examples
Understanding the relationship between volume and amount of gas is essential in various fields, such as chemistry and engineering. For instance, this concept helps in determining the amount of air required to fill tires to a specific pressure at different temperatures or calculating how much carbon dioxide is released from a pressurized canister during usage. Applying this proportionality ensures safety and precision in handling gases in real-world equipment.