Question

The Boltzmann entropy equation is (a) \(S=\frac{K}{\ln W}\) (b) \(S=K^{r} \ln \frac{P_{2}}{P_{\mathrm{J}}}\) (c) \(S=K^{\prime} \ln \frac{V_{\mathrm{J}}}{V_{2}}\) (d) \(S=K^{\prime \prime} \ln W\)

Short answer

The correct Boltzmann entropy equation is \(S = K^{\prime \prime} \ln W\), which corresponds to answer choice (d).

Step-by-step solution

Review the Boltzmann Entropy Equation

Boltzmann entropy equation is classified from the Boltzmann principle, which relates the entropy \(S\) of an ideal gas to the quantity \(W\), the number of microstates corresponding to a given macrostate, with \(K\) being the Boltzmann constant: \(S = K*\ln W\). Therefore, looking at the choices given:

Analyze Each Given Equation

(a) \(S=\frac{K}{\ln W}\) does not match the correct formula, because in the right equation \(K\) is multiplied with \(lnW\), not divided. (b) \(S=K^{r} \ln \frac{P_{2}}{P_{\mathrm{J}}}\) includes pressure \(P\), which is not part of the original Boltzmann entropy equation. (c) \(S=K^{\prime} \ln \frac{V_{\mathrm{J}}}{V_{2}}\) This equation includes volume \(V\), which is not part of the original Boltzmann entropy equation. (d) \(S=K^{\prime \prime} \ln W\) This equation matches the structure of the Boltzmann entropy equation. The notation \(K^{\prime \prime}\) is presumably just another name for the Boltzmann constant. However, if it represents a different value, this wouldn't be correct.

Key concepts

Entropy

Entropy is a measure of disorder or randomness in a system. It's an important concept in thermodynamics and statistical mechanics. In simple terms, it tells us how energy is distributed in a system and how likely it is for a system to be in a particular state. The higher the entropy, the more disordered the system is. For instance, a gas with atoms moving around randomly has higher entropy than a solid with atoms neatly arranged.

In mathematical terms, entropy quantifies the number of different ways a system can be arranged. This is where the Boltzmann entropy equation comes in handy. It uses the natural logarithm of the number of microstates, multiplied by the Boltzmann constant, to calculate entropy.

Understanding entropy helps in various fields, such as predicting chemical reactions, understanding refrigeration cycles, and even in coding theory for data compression. It's a fundamental concept that explains why some processes occur naturally, such as ice melting or coffee cooling down.

Microstates

Microstates are specific configurations that a system can be in, with each configuration corresponding to a particular way in which the components of the system can be arranged. For example, if you have a gas in a container, a microstate would be a specific distribution of speeds and positions of each molecule within that gas.

While a macrostate is defined by macroscopic properties like temperature and pressure, a microstate delves into the microscopic level of these properties. Each macrostate can have multiple microstates, each equally probable if the system is in equilibrium.

To better understand microstates:

• Microstates show the exact distribution of individual molecules' positions and velocities.
• Regardless of the number of particles, the greater the number of microstates, the higher the entropy of the system.
• Microstates are central to the statistical description of systems in thermodynamics, especially when using equations like the Boltzmann entropy equation.

Boltzmann Constant

The Boltzmann constant is a physical constant that relates the average kinetic energy of particles in a gas with the temperature of the gas. It's denoted by the symbol \(K\) and plays a crucial role in statistical mechanics. This constant serves as a bridge between macroscopic and microscopic physics.

The value of the Boltzmann constant is approximately \(1.38 \times 10^{-23}\) Joules per Kelvin (J/K). It is a small number due to the enormous difference between microscopic and macroscopic scales.

Key points about the Boltzmann constant include:

• It appears in the famous Boltzmann entropy equation \(S = K \ln W\), which relates the entropy \(S\) of a system to the number of microstates \(W\).
• It helps in translating the microscopic behavior of particles to the macroscopic observables in thermodynamics, like temperature and pressure.
• Understanding this constant is essential for grasping concepts like entropy and temperature at a molecular level.