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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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5:56 minutes

Problem 43

Textbook Question

2.0 mol of helium at 280℃ undergo an isobaric process in which the helium entropy increases by 35 J/K. What is the final temperature of the gas?

Verified step by step guidance

Step 1: Convert the initial temperature from Celsius to Kelvin. Use the formula:

T = T

_C+273.15. This ensures the temperature is in the correct unit for thermodynamic calculations.

Step 2: Recall the formula for entropy change in an isobaric process for an ideal gas:

ΔS = n C

_pln(T_f/T_i), where

n

is the number of moles,

C

_p is the molar heat capacity at constant pressure,

T

_i is the initial temperature, and

T

_f is the final temperature.

Step 3: For helium, a monatomic ideal gas, the molar heat capacity at constant pressure is

C

_p=5R/2, where

R

is the universal gas constant (

8.314 J / mol \cdot K

). Substitute this value into the entropy formula.

Step 4: Rearrange the formula to solve for the final temperature

T

_f:

T

_f=T_i·exp(ΔS/(nC_p)). Substitute the values for

ΔS

n

C

_p, and

T

_i into the equation.

Step 5: Perform the calculation to find the final temperature

T

_f. Ensure all units are consistent throughout the calculation (e.g., Kelvin for temperature, J/K for entropy).

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Key Concepts

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

Isobaric Process

An isobaric process is a thermodynamic process in which the pressure remains constant while the volume and temperature of the gas may change. In such processes, the heat added to the system results in work done by the system as it expands. This concept is crucial for understanding how gases behave under constant pressure conditions, particularly in relation to changes in temperature and entropy.

Entropy

Entropy is a measure of the disorder or randomness in a system, often associated with the amount of energy unavailable for doing work. In thermodynamics, an increase in entropy indicates that the system has absorbed heat and undergone a transformation towards a more disordered state. Understanding entropy is essential for analyzing energy transfers and the direction of thermodynamic processes, especially in relation to temperature changes.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and number of moles of an ideal gas. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature. This law is vital for calculating the final state of a gas after a thermodynamic process, such as determining the final temperature in the given problem.

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