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Chapter 20: Problem 31

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.

Short Answer

Expert verified
The entropy change is approximately 10.06 J/K.

Step by step solution

01

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.
02

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路K), \( V_f \) is the final volume, and \( V_i \) is the initial volume.
03

Calculate Initial and Final Volume

The initial volume \( V_i \) of the balloon is given as 2.40 L. Since 1 m鲁 = 1000 L, we convert this to 0.00240 m鲁. The final volume \( V_f \) is the volume of the entire space station, 425 m鲁.
04

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.
05

Calculate Natural Logarithm

Calculate \( \ln \left( \frac{425}{0.00240} \right) \) to get \( \ln (177083.33) \approx 12.09 \).
06

Calculate Entropy Change

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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路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.

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Most popular questions from this chapter

Compare the entropy change of the warmer water to that of the colder water during one cycle of the heat engine, assuming an ideal Carnot cycle. (a) The entropy does not change during one cycle in either case. (b) The entropy of both increases, but the entropy of the colder water increases by more because its initial temperature is lower. (c) The entropy of the warmer water decreases by more than the entropy of the colder water increases, because some of the heat removed from the warmer water goes to the work done by the engine. (d) The entropy of the warmer water decreases by the same amount that the entropy of the colder water increases.

A Carnot engine is operated between two heat reservoirs at temperatures of 520 \(K\) and 300 \(K\). (a) If the engine receives 6.45 \(kJ\) of heat energy from the reservoir at 520 \(K\) in each cycle, how many joules per cycle does it discard to the reservoir at 300 \(K\)? (b) How much mechanical work is performed by the engine during each cycle? (c) What is the thermal efficiency of the engine?

An average sleeping person metabolizes at a rate of about 80 \(W\) by digesting food or burning fat. Typically, 20% of this energy goes into bodily functions, such as cell repair, pumping blood, and other uses of mechanical energy, while the rest goes to heat. Most people get rid of all this excess heat by transferring it (by conduction and the flow of blood) to the surface of the body, where it is radiated away. The normal internal temperature of the body (where the metabolism takes place) is 37\(^\circ\)C, and the skin is typically 7 C\(^\circ\) cooler. By how much does the person's entropy change per second due to this heat transfer?

Three moles of an ideal gas undergo a reversible isothermal compression at 20.0\(^\circ\)C. During this compression, 1850 J of work is done on the gas. What is the change of entropy of the gas?

A Carnot heat engine uses a hot reservoir consisting of a large amount of boiling water and a cold reservoir consisting of a large tub of ice and water. In 5 minutes of operation, the heat rejected by the engine melts 0.0400 kg of ice. During this time, how much work \(W\) is performed by the engine?
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