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23. The Second Law of Thermodynamics

Entropy and the Second Law of Thermodynamics

Physics

23. The Second Law of Thermodynamics

Entropy and the Second Law of Thermodynamics

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6:24 minutes

Problem 51a

Textbook Question

Two samples of an ideal gas are initially at the same temperature and pressure. They are each compressed reversibly from a volume V to volume V/2, one isothermally, the other adiabatically. In which sample is the final pressure greater?

Verified step by step guidance

Understand the two processes: In an isothermal process, the temperature remains constant, and the pressure-volume relationship is governed by Boyle's Law:

P V_{1} = P_{2} V_{2}

. In an adiabatic process, no heat is exchanged with the surroundings, and the pressure-volume relationship is governed by the adiabatic equation:

P^{V} γ = 	ext{constant}

, where

γ

is the adiabatic index (ratio of specific heats).

Write the isothermal process equation: For the isothermal compression, the pressure doubles as the volume is halved. Using Boyle's Law,

P_{2} = 2 P_{1}

, where

P_{1}

is the initial pressure.

Write the adiabatic process equation: For the adiabatic compression, use the equation

P_{1} V^{γ} = P_{2} V^{γ}

. Substituting

V_{2} = V_{1} /2

, solve for

P_{2}

P_{2} = P_{1} imes 2 γ

Compare the final pressures: In the isothermal process, the final pressure is

2 P_{1}

. In the adiabatic process, the final pressure is

P_{1} imes 2 γ

. Since

γ > 1

for an ideal gas, the adiabatic process results in a higher final pressure.

Conclude: The final pressure is greater in the sample that undergoes the adiabatic compression because the temperature increases during the adiabatic process, leading to a higher pressure compared to the isothermal process where the temperature remains constant.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ideal Gas Law

The Ideal Gas Law describes the relationship between pressure, volume, temperature, and the number of moles of an ideal gas, expressed as PV = nRT. This law is fundamental in understanding how gases behave under various conditions, particularly during processes like compression. It helps predict how changes in volume and temperature affect pressure.

Isothermal Process

An isothermal process occurs at a constant temperature, meaning that any heat generated or absorbed during the compression of the gas is exchanged with the surroundings. In this scenario, when the gas is compressed isothermally from volume V to V/2, the pressure increases according to the Ideal Gas Law, but the temperature remains unchanged.

Adiabatic Process

An adiabatic process is characterized by no heat exchange with the surroundings, meaning all the work done on the gas results in a change in internal energy and temperature. When the gas is compressed adiabatically from volume V to V/2, the pressure increases significantly more than in the isothermal case because the temperature of the gas rises due to the work done on it.

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