Question

The entropy of a macroscopic state is given by \(S=k_{B} \ln w\) where \(k_{\mathrm{B}}\) is the Boltzmann constant and \(w\) is the number of possible microscopic states. Calculate the change in entropy when \(n\) moles of an ideal gas undergo free expansion to fill the entire volume of a box after a barrier between the two halves of the box is removed.

Short answer

In the given problem, we have n moles of an ideal gas undergoing a free expansion in a box. The change in entropy (ΔS) for this process is given by: ΔS = Nk_B * ln(2), where N is the number of moles and k_B is the Boltzmann constant.

Step-by-step solution

Calculate the initial microscopic states (w_initial)

In the beginning, the gas is restricted to half of the box volume, so \(V_i = \frac{V}{2}\). Using the microcanonical ensemble formula and replacing \(V_i\) with \(\frac{V}{2}\), we get: \(w_{initial} = \frac{(\frac{V}{2}^N N!)^{1/2}}{h^N (2 \pi m k_B T)^{3N/2}}\)

Calculate the initial entropy (S_initial)

Now we use the given entropy formula: \(S=k_B \ln w\) to find the initial entropy: \(S_{initial} = k_B \ln w_{initial} = k_B \ln \frac{(\frac{V}{2}^N N!)^{1/2}}{h^N (2 \pi m k_B T)^{3N/2}}\)

Calculate the final microscopic states (w_final)

After the free expansion, the gas fills the entire box volume, which is \(V\). Replacing the volume in the microcanonical ensemble formula with V, we get: \(w_{final} = \frac{(V^N N!)^{1/2}}{h^N (2 \pi m k_B T)^{3N/2}}\)

Calculate the final entropy (S_final)

Using the entropy formula and the obtained value for \(w_{final}\), we can calculate the final entropy: \(S_{final}=k_B \ln w_{final} = k_B \ln \frac{(V^N N!)^{1/2}}{h^N (2 \pi m k_B T)^{3N/2}}\)

Calculate the change in entropy (ΔS)

The change in entropy is given by \(\Delta S = S_{final} - S_{initial}\). Subtracting the initial entropy from the final entropy, we get: \(\Delta S = k_B \ln \frac{(V^N N!)^{1/2}}{h^N (2 \pi m k_B T)^{3N/2}} - k_B \ln \frac{(\frac{V}{2}^N N!)^{1/2}}{h^N (2 \pi m k_B T)^{3N/2}}\) We can simplify this expression using the logarithm properties: \(\Delta S = k_B \ln \frac{(V^N N!)^{1/2}}{(\frac{V}{2}^N N!)^{1/2}} = k_B \ln (2^N) = Nk_B \ln 2\) The change in entropy for the free expansion of n moles of an ideal gas in a box is \(\Delta S = Nk_B \ln 2\).

Key concepts

Entropy

Entropy is a measure of disorder or randomness in a system. It captures how energy is dispersed among microstates of a system. In simpler terms, it's about how spread out and mixed things are. The formula for entropy that we use is \[ S = k_B \ln w \] where:

• \( S \) is entropy
• \( k_B \) is Boltzmann's constant
• \( w \) is the number of microscopic ways the system can be arranged.

This formula tells us that as the number of microstates \( w \) increases, the entropy \( S \) of the system increases as well.
This is because there are more ways to arrange the particles without changing the macroscopic state, indicating greater disorder.
In the context of gases, entropy increases when the gas particles have more space to spread out and more possible positions to occupy.

Ideal Gas

An ideal gas is a hypothetical gas that follows the ideal gas law perfectly. The ideal gas law is expressed as \[ PV = nRT \] where:

• \( P \) is the pressure of the gas
• \( V \) is the volume of the gas
• \( n \) is the number of moles of the gas
• \( R \) is the universal gas constant
• \( T \) is the temperature in Kelvin.

Ideal gases are assumed to have perfectly elastic collisions and no interactions between the molecules, other than these collisions.
This simplification allows us to model the behavior of gases under a wide range of conditions, making calculations much simpler and more predictable.
The model is particularly useful when studying gases at high temperatures and low pressures because under these conditions, real gases behave more like ideal gases.

Free Expansion

Free expansion is a process where a gas expands into an available volume without the presence of external pressure work. In this scenario, no work is done on or by the system, and no heat is transferred because the system is isolated.
Despite this, free expansion results in an increase in entropy. In free expansion:

• The gas particles spread out to fill a larger volume.
• Since no external work is done, there's no change in internal energy.
• The entropy increases because there is a greater number of possible positions for the particles, increasing the microstates \( w \).

This explains why, even though the energy doesn't change, the disorder, or the entropy, of the gas increases.
The free expansion of an ideal gas, as calculated, shows that the change in entropy \( \Delta S \) is given by \( Nk_B \ln 2 \), indicating increased disorder.

Boltzmann Constant

The Boltzmann constant, denoted as \( k_B \), is a fundamental physical constant that plays a crucial role in thermodynamics. It relates the average kinetic energy of particles in a gas with the temperature of the gas.
This constant bridges the macroscopic and microscopic worlds, connecting statistical mechanics and classical thermodynamics through the equation:\[ S = k_B \ln w \]In this context:

• \( k_B \) serves as a proportionality constant that scales entropy calculations appropriately.
• It provides a way to calculate entropy using the number of microstates \( w \).

Understanding \( k_B \) helps in explaining how individual particle behavior contributes to larger thermodynamic properties.
It's essential for quantitatively describing how numerous microscopic configurations translate to an observable level of disorder or order in a system.