Question

A feeling for the numbers: comparing multiplicities Boltzmann's equation for the entropy (eqn 5.29 ) tells us that the entropy difference between a gas and a liquid is given by \(S_{\text {gar }}-S_{\text {liquid }}=k_{E} \ln \frac{W_{\text {Qas }}}{W_{\text {llavid }}}\) From the macroscopic definition of entropy as \(\mathrm{d} S=\mathrm{d} Q / \mathrm{T}\) we can make an estimate of the ratios of multiplicities by noting that boiling of water takes place at fixed \(T\) at 373 K. (a) Consider a cubic centimeter of water and use the result that the heat needed to boil water (the latent heat of vaporIzation) is given by \(Q_{\text {weviration }}=40.66 \mathrm{kJ} / \mathrm{mol}\) (at \(100^{\circ} \mathrm{C}\) ) to estimate the ratio of multiplicities of water and water vapor for this number of molecules. Write your result as 10 to some power. If we think of multiplicities in terms of an ideal gas at fixed \(T,\) then $$\frac{w_{1}}{W_{2}}=\left(\frac{V_{1}}{V_{2}}\right)^{N}$$ What volume change would one need to account for the liquid/vapor multiplicity ratio? Does this make sense? (b) In the chapter we discussed the Stirling approximation and the fact that our results are incredibly tolerant of error. Let us pursue that in more detail. We have found that the typical types of multiplicities for a system like a gas are of the order of \(W \approx \exp \left(10^{25}\right) .\) Now, let us say we are off by a factor of \(10^{1000}\) in our estimate of the multiplicities, namely, \(W=10^{1000} \exp \left(10^{25}\right) .\) Show that the difference in our evaluation of the entropy is utterly negligible whether we use the first or second of these results for the multiplicity. This is the error tolerance that permits us to use the Stirling approximation so casually!

Short answer

For a given set of parameters, there is a considerable change in the ratio of multiplicities from liquid to gas states, around the order of \(10^n\). The corresponding change in the volume makes sense since the volume of the gas state is usually larger than the liquid. Regarding the Stirling approximation, the entropy calculation is found to be incredibly resistant to even large changes in multiplicity, underscoring the value of the Stirling approximation.

Step-by-step solution

Calculate Ratio of Multiplicities

Using Boltzmann's equation for entropy, solve for the ratio of multiplicities. Given \(S_{\text{gas}}-S_{\text{liquid}}=k_E \ln \frac{W_{\text{gas}}}{W_{\text{liquid}}}\), where \(k_E\) is the Boltzmann constant. Also, utilizing the macroscopic definition of entropy \(\delta S=\frac{\delta Q}{T}\), where \(\delta Q = Q_{\text{vaporization}} = 40.66 \times 10^3 \, J/mol\). The ratio \(\frac{W_{\text{gas}}}{W_{\text{liquid}}}\) becomes \(=e^{T\delta S / k_E}\) after rearranging and simplifying. This value will be of the order of \(10^n\).

Derive Volume for Changes

Now, using \(\frac{w_{1}}{w_{2}}=\left(\frac{V_{1}}{V_{2}}\right)^{N}\), one can solve for the volume change \(V_1/V_2\) as the Nth root of the multiplicity ratio \(\frac{w_{1}}{w_{2}}\). Checking if the change in volume corresponds with the shift from liquid to gas is necessary in order to confirm the results.

Calculate Difference in Entropy Evaluation

To evaluate part (b), it's necessary to understand the Stirling approximation, through which the effect of very large changes in multiplicities on the entropy can be seen to be marginal. By treating two multiplicities, \(W_1=\exp(10^{25})\) and \(W_2=10^{1000}\exp(10^{25})\), and comparing the resulting entropies, we can show the effect of error tolerance. Use Boltzmann's equation for entropy to calculate and compare the resulting entropies for these two multiplicities.

Key concepts

Boltzmann's Equation

Boltzmann's equation is a fundamental formula in statistical mechanics that links entropy to the multiplicity of a system. It expresses the idea that the more ways a system's components can be arranged, the greater the entropy. The equation is given by \( S = k_B \ln W \), where \( S \) is the entropy, \( k_B \) is the Boltzmann constant, and \( W \) is the multiplicity, or the number of possible microstates of the system.
When comparing the entropy of different phases, such as gas and liquid, Boltzmann's equation can determine the entropy difference using the formula \( S_{\text{gas}} - S_{\text{liquid}} = k_B \ln \frac{W_{\text{gas}}}{W_{\text{liquid}}} \).
This equation highlights how changes in multiplicity can significantly impact the entropy of a system, making Boltzmann's equation crucial to understanding the behavior of systems at a microscopic level.

Multiplicities

Multiplicities, denoted by \( W \), refer to the number of possible arrangements or microstates of the particles within a system at a certain energy level. In statistical mechanics, a higher multiplicity implies a more disordered system with higher entropy.
The concept of multiplicities is particularly important when examining phase changes, such as from liquid to gas. When a liquid boils, its molecules gain more freedom and can be arranged in significantly more ways compared to a liquid state, thus increasing its multiplicity.
The ratio of multiplicities for a gas to a liquid can be expressed as \( \frac{W_{\text{gas}}}{W_{\text{liquid}}} \). Understanding this ratio allows us to analyze the volume changes required during phase transitions, aligning with the ideal gas law and offering insights into thermodynamic properties like energy and temperature.

Latent Heat of Vaporization

The latent heat of vaporization is the amount of energy required to convert a substance from a liquid to a gas at its boiling point without changing its temperature. For water, this energy requirement is approximately 40.66 kJ/mol at 100°C.
Latent heat plays a critical role in meteorology, thermodynamics, and various engineering processes, as it governs the energy exchange during phase changes. In the context of the exercise, the latent heat of vaporization helps estimate the entropy difference and thus the ratio of multiplicities between liquid and gaseous states.
This concept is intertwined with both macroscopic and microscopic interpretations of thermodynamic systems, bridging the physical understanding of phase changes with the statistical mechanics perspective through entropy and multiplicities.

Stirling Approximation

The Stirling approximation is a mathematical tool used to simplify the computation of factorials, especially useful when dealing with very large numbers. It approximates the factorial of a number \( n \) as \( n! \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n \).
In statistical mechanics, this approximation proves valuable for calculating multiplicities and entropy, given the inherently large numbers involved. Even if there's a considerable variation in the estimated multiplicities, such as being off by a factor of \( 10^{1000} \), the resulting impact on entropy calculations is minimal.
This remarkable error tolerance allows the use of Stirling's approximation to estimate multiplicities without significantly compromising the accuracy of thermodynamic predictions. It supports calculations by boosting computational efficiency without sacrificing precision in practical scenarios.