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Chapter 3: Q. 3.27 (page 112)

What partial-derivative relation can you derive from the thermodynamic identity by considering a process that takes place at constant entropy? Does the resulting equation agree with what you already knew? Explain.

Short Answer

Expert verified

The relation that can be derived from the thermodynamic identity is P=-dUdVN,S.

The derived equation agrees with the first law of thermodynamics.

Step by step solution

01

Given Information

The thermodynamic identity is:

dU=TdS-PdV

The process is taking place at constant entropy.

Hence,dS=0

02

Calculation

Since the process is taking place at constant entropy, the thermodynamic identity can be written as:

dU=TdS-PdVdU=-PdVP=-dUdVN,S

This is the derived relation.

From the relation, it can be seen that the volume change is fully responsible for the energy shift. As a result, there is no heat transfer into or out of the system. Heat flows until equilibrium is reached, resulting in a rise in entropy. As a result, the equation follows the first law of thermodynamics.

03

Final answer

The derived relation is P=-dUdVN,S. It agrees with the first law of thermodynamics.

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Most popular questions from this chapter

A bit of computer memory is some physical object that can be in two different states, often interpreted as 0 and 1. A byte is eight bits, a kilobyte is 1024=210bytes, a megabyte is 1024 kilobytes, and a gigabyte is 1024 megabytes.

(a) Suppose that your computer erases or overwrites one gigabyte of memory, keeping no record of the information that was stored. Explain why this process must create a certain minimum amount of entropy, and calculate how much.

(b) If this entropy is dumped into an environment at room temperature, how much heat must come along with it? Is this amount of heat significant?

Consider an Einstein solid for which both N and q are much greater than 1. Think of each oscillator as a separate "particle."

(a) Show that the chemical potential is

role="math" localid="1646995468663" 渭=-kTlnN+qN

(b) Discuss this result in the limits N鈮�qand N鈮�q, concentrating on the question of how much Sincreases when another particle carrying no energy is added to the system. Does the formula make intuitive sense?

An ice cube (mass 30g)0掳Cis left sitting on the kitchen table, where it gradually melts. The temperature in the kitchen is 25掳C.

(a) Calculate the change in the entropy of the ice cube as it melts into water at 0掳C. (Don't worry about the fact that the volume changes somewhat.)

(b) Calculate the change in the entropy of the water (from the melted ice) as its temperature rises from 0掳Cto 25掳C.

(c) Calculate the change in the entropy of the kitchen as it gives up heat to the melting ice/water.

(d) Calculate the net change in the entropy of the universe during this process. Is the net change positive, negative, or zero? Is this what you would expect?

Consider a monatomic ideal gas that lives at a height z above sea level, so each molecule has potential energy mgzin addition to its kinetic energy.

(a) Show that the chemical potential is the same as if the gas were at sea level, plus an additional term mgz:

渭(z)=-kTlnVN2蟺mkTh23/2+mgz.

(You can derive this result from either the definition 渭=-T(鈭�S/鈭�N)U,Vor the formula 渭=(鈭�U/鈭�N)S,V.

(b) Suppose you have two chunks of helium gas, one at sea level and one at height z, each having the same temperature and volume. Assuming that they are in diffusive equilibrium, show that the number of molecules in the higher chunk is

N(z)=N(0)e-mgz/kT

in agreement with the result of Problem 1.16.

In Section 2.5 I quoted a theorem on the multiplicity of any system with only quadratic degrees of freedom: In the high-temperature limit where the number of units of energy is much larger than the number of degrees of freedom, the multiplicity of any such system is proportional to UNf/2, whereNf is the total number of degrees of freedom. Find an expression for the energy of such a system in terms of its temperature, and comment on the result. How can you tell that this formula for惟 cannot be valid when the total energy is very small?
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