Problem 63 - Chemistry: An Atoms-Focused Approach

Question

A \(150.0 \mathrm{L}\), weather balloon contains 6.1 moles of helium but loses it at a rate of \(10 \mathrm{mmol} / \mathrm{h}\). What is the volume of the balloon after \(24 \mathrm{h}\) ?

Step-by-Step Solution

Answer

Answer: The approximate volume of the weather balloon after losing helium for 24 hours is 144.2 L.

Step 1: Calculate the number of moles lost in 24 hours.

First, we need to find out how many moles of helium the balloon loses in 24 hours. We are given the rate of loss in mmol/h, which is equivalent to \(10^{-3}\) mol/h. Convert the rate of loss to moles per hour by multiplying it by \(10^{-3}\): Loss rate: \(10 \mathrm{mmol} / \mathrm{h} \times 10^{-3} \mathrm{mol} / \mathrm{mmol} = 0.010 \mathrm{mol} / \mathrm{h}\) Now, we can calculate the moles lost in 24 hours by multiplying the rate of loss by time: Moles lost: \(0.010 \mathrm{mol} / \mathrm{h} \times 24 \mathrm{h} = 0.240 \mathrm{mol}\)

Step 2: Calculate the moles of helium remaining in the balloon.

Next, we need to find out how many moles of helium are still left in the balloon after the loss. We are given the initial number of moles of helium in the balloon (6.1 moles). Subtract the moles lost to find the remaining moles of helium: Moles remaining: \(6.1 \mathrm{mol} - 0.240 \mathrm{mol} = 5.86 \mathrm{mol}\)

Step 3: Calculate the volume of the balloon using the Ideal Gas Law.

Now we can use the Ideal Gas Law to find the volume of the balloon after 24 hours, assuming the pressure and temperature remain constant. Rearrange the Ideal Gas Law formula to solve for the volume: \(V=\frac{nRT}{P}\) For this step, we need to know the values of R and P. In this case, R is a given constant: \(R = 0.0821 \mathrm{L} \cdot \mathrm{atm} / \mathrm{mol} \cdot \mathrm{K}\). Since the pressure is assumed to be constant and we are only interested in the change in volume, we can use the ratio method. Divide the final moles by the initial moles and multiply by the initial volume to get the final volume: \(V_{final} = \frac{n_{final}}{n_{initial}} \times V_{initial}\) Using the values from the previous steps, we get: \(V_{final} = \frac{5.86 \mathrm{mol}}{6.1 \mathrm{mol}} \times 150.0 \mathrm{L} \approx 144.2 \mathrm{L}\)

Step 4: Present the final answer.

After losing helium for 24 hours, the volume of the weather balloon is approximately 144.2 L.

Key Concepts

Helium Gas

Helium is a fascinating element, known as a noble gas. It's colorless, odorless, tasteless, and incredibly light. This makes it an excellent choice for filling balloons, like in our original exercise.
The properties of helium include:

• Low density: Helium is lighter than air, which is why balloons filled with helium float.
• Inertness: As a noble gas, helium doesn't react easily with other substances, making it safe for various applications.
• High thermal conductivity: Helium conducts heat efficiently, useful in various technological applications.

Understanding these properties helps explain why helium is commonly used in balloons. Despite its inertness, helium will escape over time, impacting the balloon's lift and volume.

Mole Calculations

Calculating moles is a fundamental skill in chemistry, particularly important when dealing with gases. In the original exercise, we deal with helium measured in moles, as it's a standard unit for counting particles in chemistry.
A few key points about moles:

• One mole of any substance contains Avogadro's number of particles, approximately \(6.022 \times 10^{23}\).
• When dealing with gases, the ideal gas law is often used to relate moles to other properties like volume and pressure.
• Moles allow chemists to convert between mass, volume, and molecule count conveniently.

The mole calculations in our problem involve determining the reduction in helium over time. By understanding the rate of loss in moles, we can accurately find out how much gas remains after a certain period.

Volume of Gas

The volume of gas is a dynamic property influenced by factors like temperature, pressure, and the quantity of gas (in moles). The original exercise uses the Ideal Gas Law to focus on how the volume changes.
Important insights about gas volume include:

• The Ideal Gas Law formula is \(PV = nRT\), connecting pressure \(P\), volume \(V\), moles \(n\), the gas constant \(R\), and temperature \(T\).
• If pressure and temperature are constant, the volume varies directly with the number of moles of gas.
• Decreasing the number of moles, such as through helium loss, will lead to a decrease in the balloon's volume, as reflected in this exercise.

Mastering gas volume calculations helps you understand and predict how gases behave under different conditions, which is vital for practical applications in science and industry.