Problem 31 - University Physics with Modern Physics

Question

A lonely party balloon with a volume of 2.40 \(L\) and containing 0.100 mol of air is left behind to drift in the temporarily uninhabited and depressurized International 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 m\(^3\). Calculate the entropy change of the air during the expansion.

Step-by-Step Solution

Answer

The entropy change is approximately 10.06 J/K.

Step 1: Understanding Free Expansion

In a free expansion, the process is irreversible, and the gas expands into a vacuum without exchanging heat with the surroundings. Therefore, the internal energy of the gas does not change.

Step 2: Entropy Change Formula for Free Expansion

The entropy change (\( \Delta S \)) of an ideal gas during free expansion is given by the formula: \[ \Delta S = nR \ln \frac{V_f}{V_i} \] where \( n \) is the number of moles of gas, \( R \) is the ideal gas constant (8.314 J/mol·K), \( V_f \) is the final volume, and \( V_i \) is the initial volume.

Step 3: Calculate Initial and Final Volume

The initial volume \( V_i \) of the balloon is given as 2.40 L. Since 1 m³ = 1000 L, we convert this to 0.00240 m³. The final volume \( V_f \) is the volume of the entire space station, 425 m³.

Step 4: Applying Values to Entropy Change Formula

Substitute the values into the entropy change formula: \[ \Delta S = 0.100 \times 8.314 \times \ln \left( \frac{425}{0.00240} \right) \] Calculate the natural logarithm value first.

Step 5: Calculate Natural Logarithm

Calculate \( \ln \left( \frac{425}{0.00240} \right) \) to get \( \ln (177083.33) \approx 12.09 \).

Step 6: Calculate Entropy Change

Now calculate \( \Delta S = 0.100 \times 8.314 \times 12.09 \). The calculation yields \( \Delta S \approx 10.06 \text{ J/K} \).

Key Concepts

Free Expansion

Free expansion is a fascinating process often studied in physics, especially thermodynamics. It occurs when a gas expands into a vacuum without any obstacles or resistance. Imagine a balloon bursting in an empty room; the gas molecules will race out to fill the space.

In free expansion, there is no external pressure applied. This is why it does not perform any external work during the expansion. The process is considered to be quick and energetic.

Some key facts about free expansion include:

• The internal energy of the gas remains unchanged because it does not perform work or exchange heat.
• It is an irreversible process, meaning you can't reverse it and return to the initial state.
• It results in an increase in entropy, which is a measure of disorder or randomness in the system.

Ideal Gas Law

The ideal gas law is a master key in understanding how gases behave. It's represented by the equation \( PV = nRT \), where:

• \( P \) is the pressure of the gas.
• \( V \) is the volume of the gas.
• \( n \) is the number of moles of gas.
• \( R \) is the ideal gas constant (8.314 J/mol·K).
• \( T \) is the temperature in Kelvin.

The beauty of the ideal gas law lies in its ability to link all these fundamental gas properties.

In our free expansion example, even though the gas expands and fills a much larger space, the ideal gas law helps us understand and calculate changes, especially the entropy change, due to the increased volume.

Irreversible Process

When discussing thermodynamic processes, the term "irreversible" often pops up. Irreversible processes are those that cannot return the system to its original state without new changes occurring.

Free expansion is a classic example of an irreversible process. This is because, once the gas molecules are scattered, it's impossible to force them back into their initial confined state without interference.

Key aspects of irreversible processes:

• They often involve friction, turbulence, or mixing, which are common in real-world applications.
• They lead to an increase in entropy, representing energy dispersion and disorder.
• Recovering initial conditions is practically impossible, underlining the direction of time.

International Space Station

The International Space Station (ISS) presents a unique environment for scientific experiments, including those on gas behavior. Situated in low Earth orbit, the ISS is essentially a large, vacuum-sealed laboratory.

Here, physics takes an exciting turn due to microgravity and lack of atmosphere. This means gases behave differently compared to on Earth. Our balloon in the problem utilized the vast, empty space of the ISS for free expansion.

Notable characteristics of the ISS that affect gas experiments:

• The microgravity environment alters how gases spread and mix.
• The lack of atmosphere means there's no external pressure to counteract a gas's movement.
• Diverse environments within the ISS allow for various scientific explorations in controlled conditions.

These unique characteristics make the ISS an exceptional place for studying thermodynamics in new and enlightening ways.