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Chapter 2: Q. 2.6 (page 55)

Calculate the multiplicity of an Einstein solid with 30oscillators and 30units of energy. (Do not attempt to list all the microstates.)

Short Answer

Expert verified

An Einstein solid with 30oscillators and 30energy units has a multiplicity of惟(30,30)=591322907824307

Step by step solution

01

Step1:The number of oscillators contained in an Einstein solid.

Using the general Einstein solid multiplicity formula withN=3and q=4:

惟(N,q)=q+N鈭�1q=(q+N鈭�1)!q!(N鈭�1)!惟(3,4)=4+3鈭�14=(4+3鈭�1)!4!(3鈭�1)!=15

02

Step2:The total number of microstates

So, if we have30oscillators and energy units, the number of30microstates is:

惟(30,30)=30+30鈭�130=(30+30鈭�1)!30!(30鈭�1)!=591322907824307

惟(30,30)=591322907824307

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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.

Use the Sackur-Tetrode equation to calculate the entropy of a mole of argon gas at room temperature and atmospheric pressure. Why is the entropy greater than that of a mole of helium under the same conditions?

Fill in the algebraic steps to derive the Sackur-Tetrode equation(2.49).

For a single large two-state paramagnet, the multiplicity function is very sharply peaked about N鈫�=N/2.

(a) Use Stirling's approximation to estimate the height of the peak in the multiplicity function.

(b) Use the methods of this section to derive a formula for the multiplicity function in the vicinity of the peak, in terms of x鈮�N鈫�鈭�(N/2). Check that your formula agrees with your answer to part (a) when x=0.

(c) How wide is the peak in the multiplicity function?

(d) Suppose you flip 1,000,000coins. Would you be surprised to obtain heads and 499,000 tails? Would you be surprised to obtain 510,000 heads and 490,000 tails? Explain.

How many possible arrangements are there for a deck of 52playing cards? (For simplicity, consider only the order of the cards, not whether they are turned upside-down, etc.) Suppose you start w e in the process? Express your answer both as a pure number (neglecting the factor of k) and in SI units. Is this entropy significant compared to the entropy associated with arranging thermal energy among the molecules in the cards?
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