Question

Two identical balloons are filled to the same volume, one with air and one with helium. The next day, the volume of the airfilled balloon has decreased by \(5.0 \% .\) By what percent has the volume of the helium-filled balloon decreased? (Assume that the air is four-fifths nitrogen and one-fifth oxygen and that the temperature did not change.)

Short answer

The volume of the helium-filled balloon has decreased by approximately 13.4%.

Step-by-step solution

Understanding Graham's Law of Diffusion

Graham's Law of Diffusion states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass. Since the temperature does not change, and assuming the balloons are the same size and have the same permeability, both balloons will lose gas at a rate that is dependent on the molar mass of the gases inside.

Calculating the Relative Rate of Diffusion for Helium and Air

We calculate the relative rate of diffusion for helium to air using the formula \(\frac{Rate_{He}}{Rate_{Air}} = \sqrt{\frac{M_{Air}}{M_{He}}}\). For air, we take the average molar mass considering it is composed of four-fifths nitrogen (\(M = 28 u\text{ for }N_2\)) and one-fifth oxygen (\(M = 32 u\text{ for }O_2\)). The average molar mass of air \(M_{Air} = 0.8 \times 28 + 0.2 \times 32\). For helium, the molar mass \(M_{He} = 4 u\).

Determining the Average Molar Mass of Air

Use the weighted average based on the composition of air to determine its average molar mass: \(M_{Air} = (0.8 \times 28) + (0.2 \times 32) = 22.4 + 6.4 = 28.8 u\).

Calculating the Rate Ratio

Using Graham's Law, we find the rate ratio: \(\frac{Rate_{He}}{Rate_{Air}} = \sqrt{\frac{28.8}{4}} = \sqrt{7.2} \approx 2.68\). Thus, helium diffuses out of the balloon approximately 2.68 times faster than air.

Calculating the Percentage Volume Decreased for Helium

If air has decreased by 5.0%, and helium diffuses 2.68 times faster, then the volume decrease for helium is \(5.0\% \times 2.68 = 13.4\%\).

Key concepts

Rate of Diffusion

Rate of diffusion is a measure of how rapidly a gas spreads into a surrounding space. It's essential for understanding processes in both nature and technology, such as how a smell disperses in air or how gases are exchanged in the lungs. According to Graham's Law of Diffusion, the rate at which a gas diffuses is inversely proportional to the square root of its molar mass. This principle has practical applications in industrial processes and laboratory settings, where separation of gases based on their rates of diffusion can be crucial.

In the exercise involving balloons, Graham's Law helps us predict how fast one gas will leak compared to another. Using the information provided about the air-filled balloon, we can infer the behavior of the helium-filled balloon because helium has a significantly lower molar mass. Since lighter gases like helium diffuse more quickly, the balloon with helium deflates faster than the one filled with air.

Molar Mass of Gases

Molar mass of gases plays a pivotal role in predicting their behavior under certain conditions. It is defined as the mass of one mole of a given substance and is usually expressed in units of grams per mole (g/mol). The molar mass is used to compare the weights of different gas molecules; heavier molecules will generally diffuse more slowly than lighter ones due to their increased inertia.

In our exercise scenario, understanding the molar mass of gases allows us to calculate their relative rate of diffusion using Graham's Law. As we calculated earlier, the average molar mass of air is 28.8 u, taking into account its composition of nitrogen and oxygen. This contrasts sharply with the molar mass of helium at 4 u. The large difference between the molar masses of helium and air explains why helium escapes the balloon at a much quicker rate.

Gas Permeability

Gas permeability is the property that describes how easily a gas can pass through a material. It's a critical factor in various engineering and scientific applications, including the design of membranes for gas separation and the manufacture of products like balloons with controlled gas retention. The permeability of a material is influenced by its thickness, molecular structure, and temperature, among other things.

In the context of the exercise with the two balloons, we assume equal permeability to compare the diffusion rates accurately. If the materials of the balloons were different, this would introduce a new variable that could affect the rate of escape of the gases, independent of their molar mass. Understanding the properties of materials and how they interact with different gases is vital for predicting and controlling gas diffusion in practical applications.