Question

A $15.0$ - $\mathrm{L}$ tank is filled with ${\mathrm{H}}_2$ to a pressure of $2.00\times {10}^2$ atm. How many balloons (each $2.00\mathrm{L}$ ) can be inflated to a pressure of $1.00$ atm from the tank? Assume that there is no temperature change and that the tank cannot be emptied below $1.00\mathrm{atm}$ pressure.

Short answer

$1492$ balloons can be inflated to a pressure of $1$ atm from the $15$-L tank.

Step-by-step solution

Calculate the amount of hydrogen available for inflation

To calculate the amount of hydrogen available for inflation, you need to know the initial volume and pressure of the gas in the tank and the final pressure in the tank. The final pressure of the tank cannot go below 1 atm. Apply Boyle's Law formula to find the final volume of the gas inside the tank. The initial volume and pressure of the tank are provided, $V_1=15.0L$ and $P_1=2.00\times {10}^{2}atm$, and the final pressure is $P_2=1.00atm$. $P_1{V}_1=P_2{V}_2$ $V_2=\frac{P_1{V}_1}{P_2}$ $V_2=\frac{(2.00\times {10}^{2}atm)(15.0L)}{1.00atm}$ $V_2=3.00\times {10}^{3}L$

Calculate the amount of hydrogen that can be used to fill balloons

Now that you have the final volume of the gas inside the tank with a pressure of 1 atm, you can determine the amount of hydrogen that can be used to fill the balloons. To do so, subtract the initial volume of the tank from the final volume of the gas inside the tank. Amount of hydrogen available for inflation = $V_2-V_1$ Amount of hydrogen available for inflation = $3.00\times {10}^{3}L-15.0L$ Amount of hydrogen available for inflation ≈ $2,985L$

Calculate the number of balloons that can be filled

To find the number of balloons that can be filled, divide the amount of hydrogen available for inflation by the volume of a single balloon inflated to 1 atm pressure. The volume of each balloon is given as $2.00L$. Number of balloons = $\frac{\text{amount of hydrogen available for inflation}}{\text{volume of one balloon at 1 atm}}$ Number of balloons = $\frac{2,985L}{2.00L}$ Number of balloons ≈ 1,492.5 However, you cannot have half a balloon, so the answer will be rounded down to the nearest whole number. Therefore, 1492 balloons can be inflated to a pressure of 1 atm from the 15-L tank.

Key concepts

Understanding Gas Laws

Gas laws govern the behavior of gases under various conditions of temperature, pressure, and volume. These laws are crucial for predicting how gases will respond when these conditions change. We refer to Boyle's Law, Charles's Law, Gay-Lussac's Law, and the Combined Gas Law, all of which are derived from the ideal gas law equation.

Boyle's Law, in particular, focuses on the relationship between pressure and volume, indicating that for a given mass of gas at constant temperature, the pressure of the gas is inversely proportional to its volume. Mathematically, Boyle's Law is expressed as:
$P_1{V}_1=P_2{V}_2$,
where $P1$ and $V1$ represent the initial pressure and volume, and $P2$ and $V2$ are the final pressure and volume, respectively. Understanding this law is key to many scientific and industrial applications, including calculating how much gas can be stored or transferred under changing pressures.

Boyle's Law Application

Applying Boyle's Law is straightforward once we understand the relationship between pressure and volume. Let's say a diver descends deeper into the ocean. As the depth increases, the water pressure increases, and according to Boyle's Law, the volume of air in the diver's lungs would decrease proportionally. Similarly, when filling balloons with gas, Boyle's Law helps to determine the final volume once the pressure is changed.

When dealing with calculations, it's important to keep the units consistent and remember that the product of the pressure and volume before the change must equal the product after the change, provided that the temperature remains constant. The practical application of Boyle's Law enables us to perform accurate calculations for a variety of real-life situations, including the inflation of balloons, as in the textbook exercise we're discussing.

Calculating Gas Volume

Calculating gas volume using Boyle's Law can be incredibly useful in many scenarios, such as predicting the behavior of gases under pressure or determining the capacity needed to store gases. In our exercise, the challenge was to find out how many balloons could be inflated from a tank of hydrogen at a specific pressure without a temperature change.

By rearranging Boyle's Law, we solve for the unknown volume after pressure changes:
$V_2=\frac{P_1{V}_1}{P_2}$.
Here, $V_2$ is the volume we want to find, $P_1$ and $V_1$ are the known initial pressure and volume, and $P_2$ is the final pressure. After calculating the available gas volume that can be used for inflation, we divided this by the volume needed for one balloon to find out the number of balloons that can be inflated. Such calculations are not only essential for homework problems but also for various applications in fields like engineering, environmental science, and healthcare.