Question

In statistical mechanics, we frequently use the approximation \(\ln N !=N \ln N-N\) 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 \(N=10^{23},\) and so justify this commonly used approximation.

Short answer

\( \ln N! \approx N \ln N - N \)

Step-by-step solution

Write out \ln N! using Stirling's Formula

Stirling's formula for the factorial of a large number is given by \[ N! \approx \sqrt{2 \pi N} \left( \frac{N}{e} \right)^N \]. Taking the natural logarithm of both sides, we get \[ \ln N! \approx \ln \left( \sqrt{2 \pi N} \left( \frac{N}{e} \right)^N \right). \]

Simplify the Logarithm

Using properties of logarithms, \[ \ln N! \approx \ln \left( \sqrt{2 \pi N} \right) + \ln \left( \frac{N}{e} \right)^N. \] The logarithm of a product is the sum of the logarithms: \[ \ln N! \approx \ln \left( \sqrt{2 \pi N} \right) + N \ln \left( \frac{N}{e} \right). \]

Further Simplify the Logarithms

Breaking down further, we get: \[ \ln N! \approx \ln \left( 2 \pi N \right)^{1/2} + N \ln N - N \ln e. \] Since \( \ln \left( 2 \pi N \right)^{1/2} = \frac{1}{2} \ln (2 \pi N) \) and \( \ln e = 1 \), this becomes: \[ \ln N! \approx \frac{1}{2} \ln (2 \pi N) + N \ln N - N. \]

Compute the Approximation for \(N=10^{23}\)

Let us evaluate each term for \( N = 10^{23} \): First term: \[ \frac{1}{2} \ln (2 \pi \times 10^{23}) = \frac{1}{2} (\ln 2 + \ln \pi + \ln 10^{23}) = \frac{1}{2} (\ln 2\pi + 23 \ln 10) \approx \frac{1}{2} (1.8379 + 53.02585) \approx 27.4329. \] Second term: \[ N \ln N = 10^{23} \ln 10^{23} = 10^{23} \times 23 \ln 10 \approx 10^{23} \times 52.02585 = 5.302585 \times 10^{24}. \] Third term: \[ -N = -10^{23}. \]

Combine the Terms

Summing up all computed terms, we get: \[ \ln N! \approx 27.4329 + 5.302585 \times 10^{24} - 10^{23} \approx 5.202585 \times 10^{24} \] which simplifies to: \[ \ln N! \approx N \ln N - N. \]

Key concepts

statistical mechanics

Statistical mechanics is a branch of physics that uses statistical methods to explain the behavior of systems with a large number of particles. It helps connect the microscopic properties of individual atoms and molecules to the macroscopic or bulk properties of materials that we observe in everyday life. In essence, statistical mechanics bridges the gap between the microscopic world and macroscopic observables.
For example, properties like temperature, pressure, and volume of gases can be understood through statistical mechanics by studying the energy distribution and collisions of individual gas molecules.
This branch of physics relies heavily on probabilities and averages because dealing with each particle individually would be impractical given their vast numbers.
In the context of the exercise, statistical mechanics often requires approximations like Stirling's formula to simplify complex calculations, especially when dealing with quantities as large as Avogadro's number.

factorial approximation

Factorials are products of an integer and all the positive integers below it, often denoted as N!. When N is large, calculating the factorial directly becomes computationally intensive. That’s where factorial approximation comes in handy.
One of the most useful tools for approximating factorials of large numbers is Stirling's formula. It states that for large N, the factorial of N (N!) can be approximated as follows:
\[ N! \approx \sqrt{2 \pi N} \left( \frac{N}{e} \right)^N \]
Taking the natural logarithm of both sides yields a form more convenient for many calculations:
\[ \ln N! \approx \frac{1}{2} \ln (2 \pi N) + N \ln N - N \]
This approximation simplifies many calculations in statistical mechanics, allowing for easier manipulation of equations involving large factorials.

natural logarithm

The natural logarithm, commonly denoted as \(\ln\), is a logarithm to the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828.\
It is one of the most important functions in mathematics and is widely used in various scientific fields.
One of the key properties of the natural logarithm that makes it useful is its relationship with exponents. Specifically, if \(y = e^x\), then \(\ln y = x\). This property simplifies many mathematical expressions and helps in solving equations involving exponential growth or decay.
In the exercise, converting the factorial into a natural logarithm form simplifies the application of Stirling's formula. For large values of \(N\), this approximation gets close to the true value of the logarithm of the factorial, making it extremely useful in statistical mechanics.

large numbers

In mathematical terms, large numbers are numbers that are significantly greater than those we encounter in everyday life. They often appear in fields like cosmology, quantum mechanics, and statistical mechanics.
One of the most famous large numbers is Avogadro's number, approximately \(6.022 \times 10^{23}\), representing the number of atoms or molecules in one mole of a substance. This is incredibly useful in fields such as chemistry and physics.
When dealing with large numbers, calculations can become unwieldy. Approximations like Stirling's formula can simplify operations involving very large factorials. Complex expressions and computations break down into more manageable forms, which are easier to handle both conceptually and computationally.
In the given exercise, the number \(N = 10^{23}\) is of similar magnitude to Avogadro's number, making it an excellent candidate for using Stirling's approximation. By simplifying \(\ln N!\), we can see how such approximations yield results that align closely with more cumbersome direct calculations, demonstrating their practical utility in the realm of large-scale computations.