A balloon filled with \(1.92 \mathrm{~g}\) of helium has a volume of \(12.5 \mathrm{~L}\) What is the balloon's volume after \(0.850 \mathrm{~g}\) of helium has leaked out through a small hole (assume constant pressure and temperature)?

The concept of moles is fundamental in chemistry. A mole is a standard unit in chemistry that measures the amount of substance. When dealing with gases, we use the molar mass to convert grams of a substance to moles.
In the exercise, we started with 1.92 grams of helium. The molar mass of helium is approximately 4.00 grams per mole. Using this, we convert the initial mass of helium to moles by dividing the mass by the molar mass: \( n_i = \frac{1.92 \text{ g}}{4.00 \text{ g/mol}} = 0.48 \text{ mol} \).
After some helium leaks out, we need to figure out how many grams are left. We perform another division to find the remaining moles: \( n_f = \frac{1.07 \text{ g}}{4.00 \text{ g/mol}} = 0.2675 \text{ mol} \).
Understanding moles helps us to relate the amount of gas (mass) to how it behaves in terms of volume and other properties.

In this problem, we are using the Ideal Gas Law to understand how the volume of a gas changes when the amount of gas changes, assuming constant temperature and pressure. The Ideal Gas Law is given by \( PV = nRT \).
Pressure \( P \) and temperature \( T \) are held constant, which simplifies the relationship to \( V \rightpropto n \), meaning volume is directly proportional to the number of moles.
This means if the number of moles of gas decreases, the volume of the gas also decreases in proportion. We directly use this proportionality: \( V_f = V_i \times \frac {n_f}{n_i} \). Here, \( V_i \) is the initial volume, \( n_f \) is the final number of moles, and \( n_i \) is the initial number of moles.
By plugging numbers, we discovered how much the volume changes: \( V_f = 12.5 \text{ L} \times \frac{0.2675 \text{ mol}}{0.48 \text{ mol}} = 6.963 \text{ L} \).
This shows how knowing the relationship between volume and moles can help determine changes due to leaks.

Under constant pressure and temperature, the gas laws tell us that the volume of the gas is directly proportional to the number of moles of gas present. This fundamental understanding comes from the Ideal Gas Law, \( PV = nRT \), where pressure and temperature are constants.
In simpler terms, when the amount (moles) of gas decreases, the volume also decreases if the pressure and temperature don’t change. This directly impacts our calculations and shows why keeping these conditions stable is crucial.
In the given problem, the constant conditions mean our calculations stay valid as we account for the leaked helium. The direct proportionality between volume and moles eases the computation, reinforcing why learning these constants and their impacts is essential for solving gas-related problems.
Through such problems, students grasp how theoretical constructs like the Ideal Gas Law apply practically, bolstered by the constants that dictate gas behavior.