Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 8

A body of constant heat capacity \(C_{P}\) and at a temperature \(T_{i}\) is put in contact with a reservoir at a higher temperature \(T_{f}\). The pressure remains constant while the body comes to equilibrium with the reservoir. Show that the entropy change of the universe is equal to $$ \Delta S=C_{P}[x-\ln (1+x)] $$ where \(x=-\left(T_{f}-T_{i}\right) / T_{f}\). Prove that the entropy change is positive.

Short Answer

Expert verified
Entropy change: \[C_{P}[x - \ln(1 + x)]\], always positive when \(T_{f} > T_{i}\).

Step by step solution

01

Identify the initial parameters and relationship

Given: Heat capacity of the body is constant, denoted by \(C_{P}\), initial temperature \(T_{i}\), and final temperature \(T_{f}\). The pressure remains constant while the body comes to equilibrium with the reservoir. The temperature difference \(x\) is defined as \(x = -\frac{T_{f} - T_{i}}{T_{f}}\).
02

Determine the change in entropy of the body

For a body with constant heat capacity \(C_{P}\) coming into thermal equilibrium, the change in entropy \(\Delta S_{body}\) is given by the integral: \[\Delta S_{body} = C_{P} \int_{T_{i}}^{T_{f}} \frac{dT}{T}\]\ This simplifies to: \[\Delta S_{body} = C_{P} \, \ln \left(\frac{T_{f}}{T_{i}}\right)\]
03

Calculate the change in entropy of the reservoir

The change in entropy for the reservoir \(\Delta S_{reservoir}\) can be expressed as: \[\Delta S_{reservoir} = \frac{Q}{T_{f}}\]\ Since the heat added to the body is equal to the heat lost by the reservoir, this can be rewritten as: \[\Delta S_{reservoir} = - \frac{C_{P} (T_{f} - T_{i})}{T_{f}}\]
04

Determine the total entropy change of the universe

The total entropy change of the universe \(\Delta S_{universe}\) is the sum of the entropy changes of the body and the reservoir: \[\Delta S_{universe} = \Delta S_{body} + \Delta S_{reservoir}\]\ Substitute the expressions: \[\Delta S_{universe} = C_{P} \, \ln \left(\frac{T_{f}}{T_{i}}\right) - \frac{C_{P} (T_{f} - T_{i})}{T_{f}}\]\ Using the definition \(x = -\frac{T_{f} - T_{i}}{T_{f}}\) and simplifying: \[\Delta S_{universe} = C_{P} \left[\ln\left(\frac{T_{f}}{T_{i}}\right) + x\right]\]
05

Rewriting in the desired form

Note that: \[\ln\left(\frac{T_{f}}{T_{i}}\right) = \ln\left(\frac{T_{f}}{T_{f} + T_{i} - T_{f}}\right) = \ln\left(\frac{1}{1 + x}\right) = -\ln(1 + x)\]\ So we can rewrite the total entropy change as: \[\Delta S_{universe} = C_{P} \left[ x - \ln(1 + x) \right]\]
06

Prove the positivity of the total entropy change

To show that \(\Delta S > 0\), note that for every real number \(x > -1\), the function \(C_{P}[x - \ln(1 + x)] \) is positive due to the concavity of the logarithm function, ensuring that \[x - \ln(1 + x) > 0\]. Since \(T_{f} > T_{i}\), \(x = -\frac{T_{f} - T_{i}}{T_{f}}\) will always be in the valid range.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

thermodynamics
Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. It lays the groundwork for understanding how energy is transferred in different forms and how it affects matter. The principles of thermodynamics are essential in explaining processes in everything from engines to ecosystems.
Key concepts in thermodynamics include the laws of thermodynamics, energy transfer, and the distinctions between different forms of work and heat. Thermodynamics can also explain why certain processes are irreversible.
entropy
Entropy is a measure of the disorder or randomness in a system. It provides a way to quantify the number of specific ways a thermodynamic system can be arranged, usually referred to as the 'microstates' of the system.
The formal definition is given by: \[ S = k_B \, \text{ln} \, \text{Ω} \] where \( S \) is entropy, \( k_B \) is the Boltzmann constant, and \( \text{Ω} \) is the number of microstates.
In simple terms, higher entropy means a greater degree of disorder. For example, ice (solid water) has lower entropy compared to liquid water because the molecules in ice are more structured.
heat capacity
Heat capacity is a physical property that describes how much heat a substance needs to increase its temperature. Specifically, the heat capacity at constant pressure, denoted as \( C_{P} \), is defined as the amount of heat required to raise the temperature of a body by one degree Celsius while maintaining constant pressure.
Mathematically, it's expressed as: \[ C_{P} = \left( \frac{\partial Q}{\partial T} \right)_P \] where \( Q \) is the heat added and \( T \) is the temperature.
Heat capacity helps in understanding how different substances react to heat. For instance, water has a high heat capacity, which is why it’s effective in thermal management.
second law of thermodynamics
The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time. This principle explains why certain processes are irreversible and why natural processes tend to increase disorder.
One common formulation is: \[ \Delta S_{universe} \geq 0 \] which means that the entropy change in the universe must be positive or zero.
This law has profound implications in everyday life and engineering. It explains why heat spontaneously flows from a hot object to a cold one and why you can’t create a perpetual motion machine that continuously does work without an energy input.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A current of \(10 \mathrm{~A}\) is maintained for \(1 \mathrm{~s}\) in a resistor of \(25 \Omega\) while the temperature of the resistor is kept constant at \(27^{\circ} \mathrm{C}\). (a) What is the entropy change of the resistor? (b) What is the entropy change of the universe? The same current is maintained for the same time in the same resistor, but now thermally insulated, with the same initial temperature. If the resistor has a mass of \(10 \mathrm{~g}\) and a specific heat of \(836 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}:\) (c) What is the entropy change of the resistor? (d) What is the entropy change of the universe?

A thermally insulated cylinder, closed at both ends, is fitted with a frictionless heatconducting piston that divides the cylinder into two parts. Initially, the piston is clamped in the center with I liter of air at \(300 \mathrm{~K}\) and \(2 \mathrm{~atm}\) pressure on one side and 1 liter of air at \(300 \mathrm{~K}\) at 1 atm pressure on the other side. The piston is released and reaches equilibrium in pressure and temperature at a new position. Compute the final pressure and temperature and increase of entropy if air is assumed to be the ideal gas. What irreversible process has taken place?

Calculate the entropy change of the universe as a result of the following processes: (a) A copper block of \(0.4 \mathrm{~kg}\) mass and with heat capacity at constant pressure of \(150 \mathrm{~J} / \mathrm{K}\) at \(100^{\circ} \mathrm{C}\) is placed in a lake at \(10^{\circ} \mathrm{C}\). (b) The same block at \(10^{\circ} \mathrm{C}\) is dropped from a height of \(100 \mathrm{~m}\) into the lake. (c) Two such blocks at \(100^{\circ} \mathrm{C}\) and \(0^{\circ} \mathrm{C}\) are joined together.

Prove that the slope on a TS diagram of: (a) An isochoric curve is \(T / C_{V}\). (b) An isobaric curve is \(T / C_{P}\).

The molar heat capacity at constant magnetic field of a paramagnetic solid at low temperatures varies with the temperature and field according to the relation $$ \alpha_{\mathcal{K}}=\frac{B+C \mathcal{f}^{2}}{T^{2}}+D T^{2} $$ where \(B, C\), and \(D\) are constants. What is the entropy change of \(n\) moles of material when the temperature changes from \(T_{i}\) to \(T_{f}\) while \(\mathcal{A}\) remains constant at the value \(\pi_{0} ?\)
See all solutions

Recommended explanations on Physics Textbooks

Circular Motion and Gravitation

Read Explanation

Kinematics Physics

Read Explanation

Measurements

Read Explanation

Nuclear Physics

Read Explanation

Magnetism

Read Explanation

Electromagnetism

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free