Question

You have 100 balloons of equal volume filled with a total of \(26.8 \mathrm{~g}\) helium gas at \(23.0^{\circ} \mathrm{C}\) and \(748 \mathrm{mmHg}\). The total volume of these balloons is \(168 \mathrm{~L}\). You are given 150 more balloons of the same size and \(41.8 \mathrm{~g}\) He gas. The temperature and pressure remain the same. Determine by calculation whether you will be able to fill all the balloons with the He you have available.

Short answer

Yes, you can fill all 250 balloons with the available helium.

Step-by-step solution

Calculate Moles of Helium

The molar mass of helium is approximately 4 g/mol. We first calculate the number of moles of helium in the given situations. \[ n = \frac{26.8 \, \text{g}}{4 \, \text{g/mol}} = 6.7 \, \text{moles}\] for the first set of balloons, and \[ n = \frac{41.8 \, \text{g}}{4 \, \text{g/mol}} = 10.45 \, \text{moles} \] for the additional helium.

Use Ideal Gas Law to Calculate Total Volume

We can use the ideal gas law \( PV = nRT \) to find the volume of helium we can fill with the total available moles. For the 748 mmHg, convert it to atm: \[ 748 \, \text{mmHg} \times \frac{1 \, \text{atm}}{760 \, \text{mmHg}} = 0.984 \text{ atm} \] Let's consider the universal gas constant \(R = 0.0821 \, \text{L atm/mol K}\) and convert temperature:\( T = 23.0 ^{\circ} \text{C} + 273.15 = 296.15 \, \text{K} \).Now, plug into the ideal gas law to find the volume: \[ V = \frac{(6.7 + 10.45) \, \text{mol} \times 0.0821 \, \text{L atm/mol K} \times 296.15 \, \text{K}}{0.984 \, \text{atm}} \approx 425.41 \, \text{L}\].

Find the Required Total Volume for All Balloons

You have 100 original balloons with a total volume of 168 L and 150 additional balloons each of the same size. So, the combined number of balloons is 250. Assuming each balloon has the same volume, the volume of each balloon is \( \frac{168}{100} = 1.68 \, \text{L} \). Therefore, the total volume for 250 balloons is:\[ 250 \times 1.68 = 420 \text{ L}.\]

Compare Available Volume to Required Volume

From Step 2, we have 425.41 L of helium available. From Step 3, we need 420 L to fill all the balloons. Since 425.41 L > 420 L, we can fill all the balloons.

Key concepts

Moles of gas

When dealing with gases like helium, an important concept to grasp is the idea of moles. A mole is a unit of measurement like dozen or pair, but for atoms and molecules. It represents Avogadro's number, which is approximately 6.022 x 10^23. This number is useful because it allows chemists to count particles by weighing them.

• The molar mass is how much one mole of a substance weighs. For helium, it is 4 g/mol. This means one mole of helium atoms weighs 4 grams.
• To calculate the number of moles in a given sample, you divide the mass of the sample by the molar mass. For example, with 26.8 grams of helium: \[ n = \frac{26.8 \, \text{g}}{4 \, \text{g/mol}} = 6.7 \, \text{moles} \]
• Similarly, if you have 41.8 grams of helium: \[ n = \frac{41.8 \, \text{g}}{4 \, \text{g/mol}} = 10.45 \, \text{moles} \]

By understanding how to calculate moles, you can determine the number of particles you're working with and perform further calculations, such as with the ideal gas law.

Helium

Helium is a fascinating and useful element for various applications, including filling balloons. Being a noble gas, it is colorless, odorless, and non-reactive. Its low density allows balloons filled with helium to float. Helium, in its gaseous form, behaves predictably at many conditions, which is why we can use equations like the Ideal Gas Law to predict its behavior. Here's what makes helium especially interesting:

• Helium is the second lightest element found in the universe, which is why it's perfect for filling balloons.
• It does not catch fire or react with most substances, making it very safe for carrying out experiments or filling up balloons.
• Helium conforms to the Ideal Gas Law well, meaning its behavior in terms of pressure, volume, and temperature is easier to calculate with basic formulas. The Ideal Gas Law is expressed as: \[ PV = nRT \]

Volume calculation

In the context of gases, calculating volume using the Ideal Gas Law involves using the equation:
\[ PV = nRT \]This equation relates the pressure (P), volume (V), number of moles (n), and temperature (T) of a gas with the gas constant (R).

Here's a breakdown of how to apply this:

• Convert Units: Before you use this equation, ensure all units are in the correct form. Pressure should be in atm, volume in liters, temperature in Kelvin, and ensure that R is in compatible units, here \(0.0821 \text{ L atm/mol K}\).
• Calculate Total Moles: Add up the moles of all the helium available. Here, cumulatively, they are \(6.7 + 10.45 = 17.15 \text{ moles}\).
• Calculate Volume: Plug these values into the Ideal Gas Law to find the volume helium will occupy: \[ V = \frac{nRT}{P} \]This gives a total volume of approximately 425.41 L using the provided data.

Now, using the calculated total volume, compare it to the volume needed to fill the balloons to determine if you have enough helium.