Statistical Mechanics Lecture 6

The paragraph discusses the inherent limitations of physical devices in accurately measuring the quantities they are designed for. It uses the example of a spring balance to illustrate that while these devices can measure force within a certain range, they are not perfectly accurate. The speaker also touches on how other factors such as temperature and air pressure can affect measurements, but these are typically minor and can be ignored. The focus is on establishing a relationship between measurable quantities and the forces or conditions they depend on.

This section delves into the operation of thermometers, highlighting that they measure the temperature of the environment only when in equilibrium with it. The importance of the thermometer's design, such as holding it vertically, is emphasized to ensure accurate readings. The discussion also explores the time it takes for a thermometer to equilibrate with its surroundings and the factors that can affect this, including the thermal conductivity of the materials involved.

The speaker explores the ideal gas law and the conditions under which it begins to break down. This is approached by considering the forces between gas particles and how these interactions can alter the behavior of an ideal gas. The concept of weakly interacting particles is introduced, and the potential energy between particles is incorporated into the calculations. The goal is to understand when the ideal gas approximation is valid and when the interactions between particles become significant.

The paragraph focuses on setting up a mathematical problem to calculate the partition function for a system of molecules with potential energy between them. The total energy of the system is expressed as the sum of kinetic energies and potential energies between all pairs of molecules. The integral over the potential energy is emphasized as a crucial step in the calculation, and the concept of a small parameter expansion is introduced to manage the complexity of the problem.

This section emphasizes the role of a small parameter in statistical mechanics calculations. The potential energy between molecules is considered small compared to the kinetic energy, allowing for an expansion in a power series. The integral involving the potential energy is identified as a key quantity that determines the strength of the potential's effect on the system. The units and significance of this integral are discussed in detail.

The speaker breaks down the partition function for a gas with interactions between molecules. The partition function is separated into a product of a momentum integral and a position integral. The momentum integral is recognized as the same as that for an ideal gas, while the position integral involves the potential energy between pairs of particles. The partition function is then expressed as the product of the ideal gas partition function and a correction factor.

The paragraph discusses the use of the Taylor series expansion to approximate the logarithm of a partition function that includes a small correction term. The focus is on expanding the logarithm of (1 - x) for small x, which simplifies to -x for the purposes of the calculation. The speaker then applies this to the partition function, emphasizing the importance of the small parameter in the calculation and how it allows for a manageable expansion.

The speaker calculates the energy and pressure of a gas using the partition function derived from statistical mechanics. The energy of the gas is found by differentiating the logarithm of the partition function with respect to beta (the inverse temperature). The pressure is then determined by differentiating the logarithm of the partition function with respect to volume, at a fixed temperature. The resulting expressions for energy and pressure include terms from the ideal gas and correction terms due to particle interactions.

The paragraph explores how the behavior of a gas, specifically its energy and pressure, is affected by density and temperature. The speaker explains that while the ideal gas law holds when the potential energy between particles is much smaller than their kinetic energy, deviations occur at higher densities. The discussion also touches on how the pressure of a gas is influenced by both the density and temperature, with the pressure being proportional to the density and the temperature.

The speaker discusses the role of molecular forces, such as electrostatic forces, in determining the properties of a gas. It is noted that molecular forces typically exhibit a repulsive behavior at short distances, followed by an attractive tail at longer distances. The implications of these forces on the pressure of a gas are considered, with repulsive forces leading to increased pressure when molecules are forced close together.

The paragraph emphasizes the mathematical approach to statistical mechanics, where the calculation of partition functions and energy formulas is a straightforward process, albeit one that requires careful manipulation of symbols. The speaker highlights that while the calculations are formulaic, the interpretation of the results requires insight and an understanding of the physical phenomena they represent.

The speaker introduces the concept of exact and inexact differentials in the context of thermodynamics. It is explained that not every pair of functions that can be written as differentials necessarily represent the differential of a well-defined function. The criterion for a pair of functions to be the derivatives of a function is the vanishing of their curl. The implications of this for the calculation of heat and work in thermodynamic processes are discussed.

The paragraph concludes with a discussion on the path dependency of heat and work in thermodynamics. It is shown that while the total energy change in a cyclic process is zero, the heat added or work done can depend on the specific path taken during the process. This is demonstrated through the use of a curl test, which confirms that heat is not an exact differential, meaning it is not a state function and its value depends on the process path.