Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 2: Q. 2.14 (page 62)

Write e1023in the form 10x, for somex.

Short Answer

Expert verified

The value in the form ase1023=104.343×1022.

Step by step solution

01

Step: 1 Using logarithmic function:

The natural logarithm, which has eas its base, is the most frequent exponential function in physics and mathematics due to its straightforward features. However, logarithms may be defined in terms of any other real number, and the definition is similar to that of natural numbers. The base ten logarithms are defined as follows:

10log⁡(x)=x

Taking logarithm on both sides and using ln⁡ab=bln⁡(a),we have

ln⁡10log⁡(x)=ln⁡(x)log⁡(x)ln⁡(10)=ln⁡(x)

Converting base as exponentiation form as

e1023=10x

02

Step: 2 Finding value in form:

Taking logarithm on above equation,

e1023=10xln⁡e1023=ln⁡10x

If ln⁡ea=a;ln⁡ba=aln⁡(b)so,

1023=xln⁡(10)x=1023ln⁡(10)x=4.343×1022e1023=104.343×1022.

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!

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

For an Einstein solid with four oscillators and two units of energy, represent each possible microstate as a series of dots and vertical lines, as used in the text to prove equation 2.9.

Suppose you flip 50fair coins.

(a) How many possible outcomes (microstates) are there?

(b) How many ways are there of getting exactly25heads and25tails?

(c) What is the probability of getting exactly 25heads and 25tails?

(d) What is the probability of getting exactly 30heads and 20tails?

(e) What is the probability of getting exactly 40heads and 10 tails?

(f) What is the probability of getting 50heads and no tails?

(g) Plot a graph of the probability of getting n heads, as a function of n.

This problem gives an alternative approach to estimating the width of the peak of the multiplicity function for a system of two large Einstein solids.

(a) Consider two identical Einstein solids, each with Noscillators, in thermal contact with each other. Suppose that the total number of energy units in the combined system is exactly 2N. How many different macrostates (that is, possible values for the total energy in the first solid) are there for this combined system?

(b) Use the result of Problem2.18to find an approximate expression for the total number of microstates for the combined system. (Hint: Treat the combined system as a single Einstein solid. Do not throw away factors of "large" numbers, since you will eventually be dividing two "very large" numbers that are nearly equal. Answer: 24N/8πN.)

(c) The most likely macrostate for this system is (of course) the one in which the energy is shared equally between the two solids. Use the result of Problem 2.18to find an approximate expression for the multiplicity of this macrostate. (Answer:24N/(4πN) .)

(d) You can get a rough idea of the "sharpness" of the multiplicity function by comparing your answers to parts (b) and (c). Part (c) tells you the height of the peak, while part (b) tells you the total area under the entire graph. As a very crude approximation, pretend that the peak's shape is rectangular. In this case, how wide would it be? Out of all the macrostates, what fraction have reasonably large probabilities? Evaluate this fraction numerically for the case N=1023.

Use a pocket calculator to check the accuracy of Stirling's approximation forN=50 . Also check the accuracy of equation 2.16forlnN! .

For an Einstein solid with each of the following values of N and q , list all of the possible microstates, count them, and verify formula Ω(N,q)=q+N−1q=(q+N−1)!q!(N−1)!

(a) N=3,q=4

(b)N=3,q=5

(c) N=3,q=6

(d) N=4,q=2

(e) N=4,q=3

(f) N=1,q=anything

(g) N= anything, q=1
See all solutions

Recommended explanations on Physics Textbooks

Mechanics and Materials

Read Explanation

Torque and Rotational Motion

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Physics of Motion

Read Explanation

Nuclear Physics

Read Explanation

Fields in Physics

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