Chapter 11: Q11.3P (page 554)

In statistical mechanics, we frequently use the approximationN! = N In N-N, where N is of the order of Avogadro’s number. Write out ln N! using Stirling’s formula, compute the approximate value of each term for N = 10²³ , and so justify this commonly used approximation.

Short Answer

Expert verified

The approximation is justified.

Step by step solution

Given Information

The approximation is N! =N InN- N .

Definition of the sterling’s formula.

Sterling’s formula is used to simplify formulas involving factorial.

$n! □ n^{n} e^{- n} \sqrt{2 πn}$ .

Justify the approximation.

The approximation is N! = N In N - N.

The sterling’s formula is $n! □ n^{n} e^{- n} \sqrt{2 πn}$ .

Take log on both side on the formula mentioned above.

The equation becomes as follows.

$\begin{array}{rcl} I n (n!) & = & I n (n^{n} e^{- n} \sqrt{2 πn}) (1 + \frac{1}{12 n}) \\ = & I n n^{n} + I n e^{- n} + I n \sqrt{n} + I n (1 + \frac{1}{12 n}) \\ = & n I n n - n + \frac{1}{2} I n n + I n (1 + \frac{1}{12 n}) \end{array}$

Substitute n = 10²³ in the above equation.

The equation becomes as follows.

$n I n n = 5.526 \times 10^{26} w h e r e, n = 10^{23} I n n = 55 I n (1 + \frac{1}{12 n}) \approx 0$

Hence, In(n!) = nIn n - n .

The approximation is justified.

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In statistical mechanics, we frequently use the approximationN! = N In N-N, where N is of the order of Avogadro’s number. Write out ln N! using Stirling’s formula, compute the approximate value of each term for N = 10²³ , and so justify this commonly used approximation.

Short Answer

Step by step solution

Given Information

Definition of the sterling’s formula.

Justify the approximation.

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In statistical mechanics, we frequently use the approximationN! = N In N-N, where N is of the order of Avogadro’s number. Write out ln N! using Stirling’s formula, compute the approximate value of each term for N = 1023 , and so justify this commonly used approximation.

Short Answer

Step by step solution

Given Information

Definition of the sterling’s formula.

Justify the approximation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fluids

Fundamentals of Physics

Electrostatics

Radiation

Fields in Physics

Magnetism

Study anywhere. Anytime. Across all devices.

In statistical mechanics, we frequently use the approximationN! = N In N-N, where N is of the order of Avogadro’s number. Write out ln N! using Stirling’s formula, compute the approximate value of each term for N = 10²³ , and so justify this commonly used approximation.