Question

A \(15.0\)-L tank is filled with helium gas at a pressure of \(1.00 \times 10^{2}\) atm. How many balloons (each \(2.00 \mathrm{~L}\) ) can be inflated to a pressure of \(1.00 \mathrm{~atm}\), assuming that the temperature remains constant and that the tank cannot be emptied below \(1.00\) atm?

Short answer

To find the number of 2L balloons that can be inflated, first determine the initial moles of helium gas in the tank using the Ideal Gas Law: \( n_i = \frac{P_i \times V_i}{R \times T} \). Next, find the total moles of helium gas that can be removed to maintain 1 atm pressure in the tank: \( n_\text{removed} = n_i - n_\text{left} \). Then, determine the number of moles needed to fill a single 2L balloon: \( n_\text{balloon} = \frac{P_\text{balloon} \times V_\text{balloon}}{R \times T} \). Finally, divide the total moles of helium gas that can be removed by the moles required to fill a single balloon to find the number of balloons that can be inflated: \( \text{Number of Balloons} = \frac{n_\text{removed}}{n_\text{balloon}} \). Round the result down to the nearest whole number.

Step-by-step solution

Determine the initial moles of helium gas in the tank

Using the Ideal Gas Law formula: \(PV = nRT\). We have the initial pressure, volume, and temperature of the helium gas (assuming the temperature remains constant). We can solve for the initial moles of helium gas. Given values: \(P_i = 100 ~\text{atm}\), \(V_i = 15.0 ~\text{L}\), and \(R = 0.08206 \frac{\text{L . atm}}{{\text{mol . K}}}\). Since the temperature is constant, we can express the number of moles as follows: \[ n_i = \frac{P_i \times V_i}{R \times T} \]

Find the total moles that can be removed to keep the pressure at 1 atm

As the tank cannot be emptied below 1 atm, we need to find the total amount of moles of helium gas that can be removed to maintain this pressure. Using the Ideal Gas Law again, the number of moles left in the tank will be: \[ n_\text{left} = \frac{P_\text{left} \times V_i}{R \times T} \] Where \(P_\text{left} = 1 ~\text{atm}\). Now, to find the total moles that can be removed, subtract the number of moles left from the initial moles of helium gas: \[ n_\text{removed} = n_i - n_\text{left} \]

Find the number of moles needed to fill a single 2L balloon

To find the number of moles of helium gas needed to fill a single 2L balloon at 1 atm pressure, we can use the Ideal Gas Law again. Now, given values: \(P_\text{balloon} = 1 ~\text{atm}\), \(V_\text{balloon} = 2.00 ~\text{L}\), and \(R = 0.08206 \frac{\text{L . atm}}{{\text{mol . K}}}\). We can express the number of moles needed to fill the single balloon as: \[ n_\text{balloon} = \frac{P_\text{balloon} \times V_\text{balloon}}{R \times T} \]

Calculate the number of balloons that can be inflated

Finally, to find the number of balloons that can be inflated, divide the total moles of helium gas that can be removed by the moles required to fill a single 2L balloon: \[ \text{Number of Balloons} = \frac{n_\text{removed}}{n_\text{balloon}} \] Round the result down to the nearest whole number, as only complete balloons can be filled. This will give the total number of balloons that can be inflated under the given conditions.

Key concepts

Gas Pressure Calculation

Gas pressure plays a critical role in understanding how gases behave in different conditions. It is defined as the force that a gas exerts on the walls of its container. When dealing with the Ideal Gas Law, the pressure of gas is denoted by the letter \( P \) and is measured in units like atmospheres (atm), Pascals (Pa), or other similar units.
Knowing how to calculate gas pressure is very important in various scientific and engineering fields. Using the formula \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin, we can determine different unknown variables, including pressure.
- **Scenario Example**: If a gas occupies a volume of \(15.0L\) and is at a pressure of \(100\) atm in a confined space, this information is used to calculate other variables.
Understanding pressure helps us predict how gases will move and be utilized in practical situations, such as inflating balloons. It is fundamental to gas-related calculations and industries.

Moles of Gas

Moles of gas represent the quantity of gas present in a system. In chemistry, the term 'mole' is used to describe a specific number of particles, similar to how a dozen represents twelve items. Specifically, one mole of any substance contains \(6.022 \times 10^{23}\) particles, known as Avogadro’s number.
To find the number of moles of gas in a given condition, we use the Ideal Gas Law: \( n = \frac{PV}{RT} \). Here, \( n \) is the number of moles, \( P \) is pressure, \( V \) is volume, \( R \) is the ideal gas constant, and \( T \) is temperature. This formula allows us to calculate the moles when other factors like pressure and volume are known.
- **Practical Usage**: Consider a helium tank initially at \(100\) atm with a volume of \(15.0L\). Using the Ideal Gas Law, you can determine how many moles of helium are initially in the tank. Then, by keeping a stable pressure (e.g., not dropping below \(1\) atm), you calculate the amount of helium that can be used by finding the difference between initial and final moles.
Recognizing the number of moles helps in making predictions about gas behavior, such as determining how many balloons can be filled with a specific amount of helium, which is crucial for planning and efficient resource usage.

Gas Volume

Gas volume is a critical measurement when studying gases, especially when using the Ideal Gas Law. Volume is simply the amount of space that a gas occupies, and is most commonly measured in liters (L) or cubic meters (m³).
In the context of the Ideal Gas Law, the volume of a gas can be written within the equation as \( V \). Gas volume is influenced by other factors such as the moles of gas, temperature, and pressure, so understanding their relationship is essential.
- **Example in Practice**: Suppose you have a helium tank with a volume of \(15.0L\) that's used to fill balloons. Each balloon has a volume of \(2.0L\). The volume of the gas needs to be appropriately measured to ensure enough gas is available to fill a set number of balloons.
It is important to remember that the volume of a gas changes with temperature and pressure, as captured by the Ideal Gas Law equations. Assuring that the temperature remains constant simplifies calculations, as you only need to account for the volume ratios directly influenced by pressure changes.