Question

A helium tank contains 25.2 L of helium at 8.40 atm pressure. Determine how many 1.50 -L balloons at \(755 \mathrm{mm}\) Hg can be inflated with the gas in the tank, assuming that the tank will also have to contain He at \(755 \mathrm{mm}\) Hg after the balloons are filled (that is, it is not possible to empty the tank completely). The temperature is \(25^{\circ} \mathrm{C}\) in all cases.

Short answer

To find the number of 1.50 L balloons that can be inflated with 25.2 L of helium at 8.40 atm pressure, we first calculate the initial number of moles of helium in the tank (\(n_1\)) using the ideal gas law: \(n_1 = 9.215\, mol\). After inflating the balloons, the number of moles of helium left in the tank (\(n_2\)) is 8.610 mol. The number of moles used to fill the balloons is \(n_{used} = n_1 - n_2 = 0.605\, mol\). Each balloon requires 0.00611 mol of helium, so we can fill \(N = \frac{n_{used}}{n_{balloon}} = 99.01\) balloons. Therefore, we can fill 99 balloons with the helium in the tank.

Step-by-step solution

Convert initial conditions to SI units

Before we begin the calculations, let's convert the given values to SI units. The pressure should be in Pascals (Pa), the volume should be in cubic meters (m³), and the temperature should be in Kelvin (K). Initial pressure: \(8.40\, atm = 8.40 * 101325\,Pa = 850830\,Pa\) Initial volume: \(25.2\,L = 25.2 * 10^{-3}\,m³\) The ambient pressure is given in mm Hg, so we must convert it to pascals as well: Ambient pressure: \(755\,mm\, Hg = \frac{755}{760} * 101325\,Pa = 101000\,Pa\) Temperature is constant in all cases: \(25°C = 25 + 273.15 = 298.15\,K\)

Calculate the initial number of moles of helium

Using the ideal gas law, let's find the initial number of moles of helium in the tank. The ideal gas law equation can be written as: \[PV = nRT\] \[n = \frac{PV}{RT}\] In this equation, P = initial pressure (850830 Pa) V = initial volume (0.0252 m³) R = ideal gas constant (8.314 J/(mol K)) T = temperature (298.15 K) Now, we can calculate the initial number of moles \(n_1\): \[n_1 = \frac{850830 * 0.0252}{8.314 * 298.15} = 9.215\, mol\]

Calculate the number of moles of helium left in the tank

After helium is used to fill the balloons, the pressure in the tank will be equal to the ambient pressure (101000 Pa). We will now calculate the number of moles of helium remaining in the tank at the final conditions. Let the final volume of the tank be V2. We can use the ideal gas law to find the number of moles of helium left in the tank: \[n_2 = \frac{PV_2}{RT}\] In this equation, P = ambient pressure (101000 Pa) V2 = final volume (0.0252 m³) R = ideal gas constant (8.314 J/(mol K)) T = temperature (298.15 K) Now, we can calculate the number of moles of helium left, \(n_2\): \[n_2 = \frac{101000 * 0.0252}{8.314 * 298.15} = 8.610\, mol\]

Calculate the number of balloons that can be filled

Now that we have the initial and final number of moles of helium, we can calculate the number of moles used to fill the balloons (\(n_{used} = n_1 - n_2\)). \[n_{used} = 9.215 - 8.610 = 0.605\, mol\] Next, we need to find the number of moles required to fill a single balloon. Given that each balloon has a volume of 1.50 L and pressure of 755 mm Hg (101,000 Pa), we can use the ideal gas law to find the number of moles required: \[n_{balloon} = \frac{101000 \times (1.5 * 10^{-3})}{8.314 \times 298.15} = 0.00611\, mol\] Now, we can calculate the number of balloons that can be filled with the helium from the tank (\(N = \frac{n_{used}}{n_{balloon}}\)): \[N = \frac{0.605}{0.00611} = 99.01\] Since we cannot fill a fraction of a balloon, we round down to the nearest whole number, obtaining that we can fill 99 balloons with the helium in the tank.

Key concepts

Gas Pressure Conversion

Gas pressure conversion is essential in various scientific calculations, including those related to the Ideal Gas Law. The pressure of a gas can be expressed in different units such as atmospheres (atm), Pascals (Pa), millimeters of mercury (mm Hg), and torrs. It's important to convert gas pressure into consistent units to accurately apply the Ideal Gas Law.

To convert from atmospheres to Pascals, the conversion factor is 101325 Pa/atm. For instance, 1 atm is equivalent to 101325 Pa. When converting mm Hg to Pascals, we use the fact that 760 mm Hg is equal to 101325 Pa. By using these conversions, we ensure that all pressure readings are in the standard SI unit of pressure, which is Pascal, facilitating consistent and error-free calculations.

Molar Volume of Gas

The molar volume of a gas is the volume occupied by one mole of the gas at defined conditions of temperature and pressure. According to the Ideal Gas Law, at standard temperature and pressure (STP), which is defined as 0 degrees Celsius and 1 atmosphere, one mole of an ideal gas occupies 22.4 liters.

Understanding molar volume is crucial in stoichiometry calculations involving gases, as it allows us to predict the volume of gas produced or consumed in a chemical reaction. In practical scenarios, gases might not be at STP, and the Ideal Gas Law equation, \(PV = nRT\), allows us to calculate the volume that a certain number of moles of gas will occupy under non-standard conditions.

Stoichiometry of Gas

Stoichiometry of gas involves the calculation of the volumes, moles, or masses of reactants and products in a chemical reaction involving gases. Utilizing the Ideal Gas Law, \(PV = nRT\), we can relate the pressure, volume, and temperature of a gas to the amount in moles, providing a clear picture of the stoichiometric relationships in gas-phase reactions.

For a given gas reaction, once you know the number of moles and the balanced chemical equation, you can determine the volume of gas produced or required. This is particularly useful in practical applications such as determining how much gas is needed to inflate a balloon, as highlighted in our example problem.

SI Units Conversion

SI units conversion refers to converting different measurement units into the International System of Units (SI), which is the standard set of units for science and engineering. Common conversions to SI units include converting volume from liters to cubic meters (\(1 L = 10^{-3} m^3\)) and temperature from degrees Celsius to Kelvin (\(K = ^\circ C + 273.15\)).

Performing these conversions allows for the uniform application of scientific laws and formulas, such as the Ideal Gas Law. In the provided exercise, the correct SI unit conversion is the foundation for accurate calculations, ensuring that the problem is solved correctly.