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Chapter 12: Q22P (page 508)

Consider an object containing 6 one-dimensional oscillators (this object could represent a model of 2 atoms in an Einstein solid). There are 4 quanta of vibrational energy in the object. (a) How many microstates are there, all with the same energy? (b) If you examined a collection of 48,000 objects of this kind, each containing 4 quanta of energy, about how many of these objects would you expect to find in the microstate 000004?

Short Answer

Expert verified

number of microstates = 126

number of objects = 380.95

Step by step solution

01

Given data

Number ofID oscillators (N) = 6

Quanta (n) = 4

02

(a) Determine the number of microstates

The no. of microstates,

Ω=(n+N-1)!n!(N-1)!   =(4+6-1)!4!(6-1)!   =126

03

(b)Determine the number of objects

The no. of objects is given by,

n1=collectionofobjectsΩ =48,000126 =380.95

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Most popular questions from this chapter

In order to calculate the number of ways of arranging a given amount of energy in a tiny block of copper, the block is modeled as containing 8.7×105independent oscillators. How many atoms are in the copper block?

The entropy S of a certain object (not an Einstein solid) is the following function of the internal energy E:S=bE1/2, where b is a constant. (a) Determine the internal energy of this object as a function of the temperature.

(b) What is the specific heat of this object as a function of the temperature?

This question follows the entire chain of reasoning involved in determining the specific heat of an Einstein solid. Start with two metal blocks, one consisting of one mole of aluminum (27 g) and the other of one mole of lead (207 g), both initially at a temperature very near absolute zero (0 K). From measurements of Young’s modulus one finds that the effective stiffness of the interatomic bond modeled as a spring is 16N/mfor aluminum and 5 N/m for lead. (a) Is the number of quantized oscillators in the aluminum block greater, smaller, or the same as the number in the lead block? (b) What is the initial entropy of each block? (c) In which metal is the energy spacing of the quantized harmonic oscillators larger? (d) If we add 1 J of energy to each block, which metal now has the larger number of energy quanta? (e) In which block is the number of possible ways of arranging this of energy greater? (f) Which block now has the larger entropy? (g) Which block experienced a greater entropy change? (h) Which block experienced the larger temperature change? (i) Which metal has the larger specific heat at low temperatures? (j) Does your conclusion agree with the actual data given in Figure 12.33? (The numerical data are given in a table accompanying Problem P64.)

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 5!/[5!0!]=1, 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 expression5!/[N!5-N!]to 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.

At room temperature, show that KBT≈1/40eV. It is useful to memorize this result, because it tells a lot about what phenomena are likely to occur at room temperature.
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