Problem 36 - University Physics with Modern Physics

Question

20.36 . A lonely party balloon with a volume of 2.40 \(\mathrm{L}\) and containing 0.100 \(\mathrm{mol}\) of air is left behind to drift in the temporarily uninhabited and depressurized Intermational Space Station. Sunlight coming through a porthole heats and explodes the balloon, causing the air in it to undergo a free expansion into the empty station, whose total volume is 425 \(\mathrm{m}^{3}\) . Calculate the entropy change of the air during the expansion.

Step-by-Step Solution

Answer

The entropy change is approximately 10.04 J/K.

Step 1: Identify key variables

We are given the initial volume of the balloon as \( V_i = 2.40 \text{ L} = 0.00240 \text{ m}^3 \), since 1 \, \text{L} = 0.001 \, \text{m}^3. The final volume after expansion is given as \( V_f = 425 \, \text{ m}^3 \). The number of moles of gas in the balloon is \( n = 0.100 \, \text{mol} \).

Step 2: Use the entropy change for free expansion

In a free expansion, the external pressure is 0, so the energy transferred as work is also 0. For an ideal gas, the entropy change for a free expansion can be calculated using the equation:\[\Delta S = nR \ln\left(\frac{V_f}{V_i}\right)\]where \( R = 8.314 \, \text{J/(mol K)} \) is the ideal gas constant.

Step 3: Calculate the volume ratio

Calculate the ratio \( \frac{V_f}{V_i} \) as follows:\[\frac{V_f}{V_i} = \frac{425}{0.00240} = 177083.33\]

Step 4: Compute the entropy change

Substitute the values into the entropy change formula:\[\Delta S = 0.100 \, \text{mol} \times 8.314 \, \text{J/(mol K)} \times \ln(177083.33)\]Calculate the natural logarithm and then the entropy change:\[\ln(177083.33) \approx 12.084\]\[\Delta S = 0.100 \, \text{mol} \times 8.314 \, \text{J/(mol K)} \times 12.084 \approx 10.04 \, \text{J/K}\]

Step 5: Conclusion

The entropy change of the air in the balloon during the free expansion is calculated to be approximately \( 10.04 \, \text{J/K} \).

Key Concepts

Ideal Gas Law

The ideal gas law is a fundamental concept in chemistry and physics. It relates the pressure, volume, temperature, and amount of a gas. The formula is expressed as \( PV = nRT \), where:

• \( P \) stands for the pressure of the gas.
• \( V \) is the volume of the gas.
• \( n \) represents the number of moles of the gas.
• \( R \) is the ideal gas constant, equal to 8.314 J/(mol K).
• \( T \) is the temperature in Kelvins.

This equation helps us predict how gases will react under different conditions. When applied in calculations, it assumes the gas behaves ideally, which means particles do not interact with each other and occupy no volume.
Although real gases may deviate slightly under high pressures or low temperatures, the ideal gas law provides a close approximation.It's a useful tool for understanding gas behavior, especially in scenarios like this problem, involving an air balloon inside the International Space Station.

Free Expansion

Free expansion is a process where a gas expands into a vacuum or an area of no pressure, without doing work on the environment. In this scenario, the gas doesn't exert force against any opposing forces or pressures.
Due to this lack of external pressure, free expansion doesn't involve energy transfer as work. The initial energy of the gas is solely retained within its particles, and only the volume changes.
This makes free expansion a key concept in thermodynamics, especially when analyzing entropy changes. The increased volume causes increased disorder among particles, which corresponds to higher entropy. The entropy change in this type of process for an ideal gas can be calculated using the formula:\[\Delta S = nR \ln\left(\frac{V_f}{V_i}\right)\]Here, the entropy change \( \Delta S \) arises due to the multiplication factor involving the ratio between final and initial volumes, \( V_f \) and \( V_i \). Understanding free expansion is essential for predicting entropy changes in thermodynamic transformations without external influences.

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat, work, temperature, and energy. It defines how these elements interact in a system.
Key laws govern thermodynamics, including:

• The First Law, which states that energy cannot be created or destroyed, only transformed.
• The Second Law, which introduces the concept of entropy, indicating that in any energy transfer, some energy becomes unavailable to do work.
• The Third Law, which suggests that as temperature approaches absolute zero, the entropy of a perfect crystal approaches a constant minimum.

In the case of the balloon in the exercise, thermodynamics helps us understand the change in entropy as the gas undergoes free expansion. This is guided by the principle that even in seemingly chaotic processes, nature trends towards equilibrium and increased entropy.
Thermodynamics provides a framework for understanding how energy transformations affect the state and properties of matter, making it crucial for solving problems related to changes in energy, such as the explosive expansion described.