A balloon containing helium gas expands from 230 mL to 860 mL as more helium is added. What was the initial quantity of helium present if the expanded balloon contains $3.8 \times 10^{-4} mol$, assuming constant temperature and pressure?
Use the formula: $\frac{V_1}{n_1}=\frac{V_2}{n_2}$
A. $1.0 \times 10^{-4} mol^6$
B. $1.5 \times 10^{-4} mol$
C. $1.0 \times 10^{-3} mol$
D. $1.4 \times 10^{-3} mol$
Answer (1)
Use the formula n 1 V 1 = n 2 V 2 .
Rearrange the formula to solve for n 1 : n 1 = V 2 V 1 n 2 .
Substitute the given values: n 1 = 860 mL ( 230 mL ) ( 3.8 × 1 0 − 4 mol ) .
Calculate the initial quantity of helium: n 1 = 1.016279 × 1 0 − 4 mol ≈ 1.0 × 1 0 − 4 mol .
The initial quantity of helium is 1.0 × 1 0 − 4 m o l .
Explanation
Problem Analysis We are given the initial and final volumes of a balloon containing helium gas, as well as the final quantity of helium. We are asked to find the initial quantity of helium, assuming constant temperature and pressure. We can use the formula n 1 V 1 = n 2 V 2 , where V 1 and n 1 are the initial volume and quantity of helium, and V 2 and n 2 are the final volume and quantity of helium.
Given Information We are given:
Initial volume, V 1 = 230 mL Final volume, V 2 = 860 mL Final quantity, n 2 = 3.8 × 1 0 − 4 mol
We need to find the initial quantity, n 1 .
Rearranging the Formula We can rearrange the formula to solve for n 1 :
n 1 V 1 = n 2 V 2 n 1 = V 2 V 1 n 2
Substituting Values Now, we substitute the given values into the formula: n 1 = 860 mL ( 230 mL ) ( 3.8 × 1 0 − 4 mol ) n 1 = 860 230 × 3.8 × 1 0 − 4 mol n 1 = 860 874 × 1 0 − 4 mol n 1 = 1.016279 × 1 0 − 4 mol
Final Answer Therefore, the initial quantity of helium present in the balloon was approximately 1.016279 × 1 0 − 4 mol. Among the given options, 1.0 × 1 0 − 4 m o l is the closest to our calculated value.
Examples
Understanding the relationship between volume and the amount of gas is crucial in various real-world applications. For instance, in scuba diving, knowing how the volume of air in a tank changes with the amount of air consumed helps divers manage their air supply and plan their dives safely. Similarly, in weather forecasting, understanding how the volume of air masses changes with temperature and pressure is essential for predicting weather patterns. In the medical field, ventilators rely on precise control of gas volumes to assist patients with breathing. The principles applied here help ensure accurate and safe delivery of gases in these and many other scenarios.
