Learning Materials
Features
Discover
Chapter 12: Q19Q (page 508)
A carbon nanoparticle (very small particle) contains 6000 carbon atoms. According to the Einstein model of a solid, how many oscillators are in this block?
The number of oscillators in the block is
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
At room temperature (293 K), calculate in joules and eV.
A microscopic oscillator has its first and second excited states and above the ground-state energy. Calculate the Boltzmann factor for the ground state, first excited state, and second excited state, at room temperature.
Explain why it is a disadvantage for some purposes that the specific heat of all materials decreases a low temperature.
The reasoning developed for counting microstates applies to many other situations involving probability. For example, if you flip a coin 5 times, how many different sequences of 3 heads and 2 tails are possible? Answer: 10 different sequences, such as HTHHT or TTHHH. In contrast, how many different sequences of 5 heads and 0 tails are possible? Obviously only one, HHHHH, and our equation gives , using the standard definition that 0! is defined to equal 1.
If the coin is equally likely on a single throw to come up heads or tails, any specific sequence like HTHHT or HHHHH is equally likely. However, there is only one way to get HHHHH, while there are 10 ways to get 3 heads and 2 tails, so this is 10times more probable than getting all heads. Use the expressionto calculate the number of ways to get 0 heads, 1 head, 2 heads, 3 heads, 4 heads, or 5 heads in a sequence of 5 coin tosses. Make a graph of the number of ways vs. the number of heads.
Verify that this equation gives the correct number of ways to arrange 0, 1, 2, 3, or 4 quanta among 3 one-dimensional oscillators, given in earlier tables (1, 3, 6, 10, 15).
Q1 | Q2 | #ways1 | #ways2 | #ways1 #ways2 |
0 | 4 | 1 | 15 | 15 |
1
| 3
| 3
| 10 | 30 |
2 | 2
| 6
| 6
| 36 |
3
| 1
| 10
| 3
| 30 |
4
| 0
| 15
| 1
| 15 |
What do you think about this solution?
We value your feedback to improve our textbook solutions.
Explanations 
Textbooks 
Exams 
GCSE 
IGCSE 
AS 
A Level 
International A Level 
University Admissions Tests 
Flashcards 
Vaia AI 
Notes 
Study Plans 
Study Sets 
Find a job 
Find a degree 
Student Deals 
Magazine 
Mobile App