Problem 84 - Chemistry: An Atoms-Focused Approach

Question

Four empty balloons, each with a mass of \(10.0 \mathrm{g},\) are inflated to a volume of \(20.0 \mathrm{L}\). The first balloon contains He, the second Ne, the third \(\mathrm{CO}_{2},\) and the fourth CO. If the density of air at \(25^{\circ} \mathrm{C}\) and 1.00 atm is \(1.17 \mathrm{g} / \mathrm{L},\) how many of the balloons float in it?

Step-by-Step Solution

Answer

Answer: _____ balloons

Step 1: Calculate Mass of Each Gas

We first need to find the number of moles of each gas in the balloon using the Ideal Gas Law equation: PV = nRT Where: P = Pressure (1.00 atm) V = Volume (20.0 L) n = Number of moles R = Gas constant (0.0821 \(atm \cdot L^{−1} \cdot K^{−1} \cdot mol^{−1}\)) T = Temperature (25°C in Kelvin = 298K) We can solve the equation for n: n = \(\frac{PV}{RT}\) Now we can calculate the number of moles for each gas in the balloons.

Step 2: Determine Mass of Each Gas in the Balloons

Using the molar mass of each gas, we can calculate the mass of each gas inside the balloons. The molar mass of He is 4.00 g/mol, Ne is 20.2 g/mol, \(\mathrm{CO}_{2}\) is 44.0 g/mol, and CO is 28.0 g/mol. The mass of a gas in a balloon (m_gas) can be determined using: m_gas = n × M_gas Where: n = Number of moles in the balloon M_gas = Molar mass of the gas

Step 3: Calculate Effective Mass of Each Balloon

The effective mass of each balloon (m_eff) can be calculated as: m_eff = mass of the balloon + mass of the gas - mass of the displaced air Where: mass of the balloon = 10.0 g mass of the gas = m_gas (calculated in Step 2) mass of the displaced air = density of air × volume of the balloon = 1.17 g/L × 20 L

Step 4: Determine if the Balloons Float

To determine if each balloon will float in air, we need to compare the effective mass of each balloon with the mass of the displaced air. If the effective mass of a balloon is less than the mass of displaced air, then the balloon will float. Otherwise, it will not float. Calculate the effective mass of each balloon and compare it to the mass of displaced air. If m_eff < mass of displaced air, then the balloon will float. Count the number of balloons that meet this condition to find how many balloons float in air.

Key Concepts

Balloon Buoyancy

Understanding why a balloon floats is all about buoyancy, which is the ability of something to float in a fluid. In our case, the fluid is air. The science behind buoyancy is based on Archimedes' principle, which states that the buoyant force on an object is equal to the weight of the fluid that the object displaces. For a balloon to float, the upward force (buoyant force) from displaced air must be greater than the total weight of the balloon, which includes the balloon's own weight and any gas inside.
The key factors in determining buoyancy include:

• The weight of the balloon and its contents.
• The density of the air.
• The volume of air displaced by the balloon.

When the mass of a balloon is less than the mass of the air it displaces, the buoyant force will lift the balloon, causing it to float. Hence, lighter gases like helium can help balloons achieve floatation because they result in a lower overall balloon mass.

Gas Density

Gas density refers to how much mass is contained in a given volume of gas. When we talk about gases like helium, neon, carbon dioxide, or carbon monoxide, each has a different density because their respective atoms or molecules vary in size and weight. The density of any gas is affected by its molar mass and the conditions it is under (such as pressure and temperature).
In general terms, density can be calculated using the formula:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]In our balloon problem, we know the density of air, and we use this information to find out if the balloons in question will float. The trick lies in comparing the mass of the gas inside each balloon to the air it displaces. If the balloon and gas together weigh less than the displaced air, the density is effectively less, which allows the balloon to float. This makes lighter gases such as helium preferable for filling balloons meant to float.

Molar Mass Calculations

Molar mass is a key concept in chemistry that refers to the mass of one mole of a given substance, usually expressed in units of grams per mole (g/mol). It directly influences the behavior of gases under the same conditions, principally affecting their density. Molar mass calculations are essential when determining how many moles of a gas are present based on a known volume, temperature, and pressure. The Ideal Gas Law, represented by the equation:\[ PV = nRT \]is an important tool in this calculation, where:

• \(P\) is pressure,
• \(V\) is volume,
• \(n\) is the number of moles,
• \(R\) is the gas constant, and
• \(T\) is temperature.

With the number of moles \(n\) found from the Ideal Gas Law, one can then calculate the mass of the gas using its molar mass:\[ \text{Mass} = n \times \text{Molar Mass} \]This is crucial in understanding how much a balloon weighs when filled with a certain gas, helping determine whether it will float in an environment like air.