22. The Boltzmann Constant and First Law of Thermodynamics

The paragraph discusses the concept of temperature and its quantitative measurement using the absolute Kelvin scale. It explains the process of finding the Kelvin scale using a gas at low concentration inside a piston and cylinder, and how the product of pressure (P) times volume (V) is a straight line for any gas, leading to the understanding that pressure times volume is linearly proportional to temperature. The paragraph also touches on the historical view of heat as a separate entity and introduces the idea of conservation of heat energy, suggesting a relationship between mechanical and heat energy.

This paragraph delves into James Prescott Joule's experiment, which determined the conversion ratio of mechanical energy to heat energy. It describes Joule's apparatus and the principle behind the experiment, which involves letting a weight fall and measuring the heat generated in water due to the loss of mechanical energy. The experiment established that 4.2 joules of mechanical energy is equivalent to 1 calorie. The paragraph also explores the microscopic understanding of heat, questioning what heat truly is and setting the stage for a deeper discussion on the nature of heat and energy storage.

The focus of this paragraph is on the proportionality constant in the equation PV=CT, where P is pressure, V is volume, and T is temperature. It discusses the historical understanding of this constant and how it was found to vary depending on the gas used. The paragraph explains that the constant is related to the 'amount of gas' and how this led to the realization that the mass of the gas must be divided by a specific number for different gases to find the effective mass in terms of pressure. This discussion paves the way for the introduction of the Boltzmann Constant and the concept of Avogadro's Number.

This paragraph introduces the Boltzmann Constant, a universal constant that appears in the proportionality of pressure, volume, and temperature. It explains how the Boltzmann Constant is derived from the masses of atoms and molecules, and how it leads to the concept of Avogadro's Number, which is used to define a mole. The paragraph emphasizes the importance of these constants in understanding the microscopic basis of the PV=CT equation and the relationship between the macroscopic properties of gases and the behavior of their constituent atoms and molecules.

The paragraph explores the microscopic basis for pressure in a gas, using a cube of gas as an example. It explains how the pressure exerted on the walls of the cube is due to the collisions of gas molecules with the walls. The paragraph simplifies the complex motion of molecules by assuming that one-third of them move in each primary direction and that all have the same speed. It then calculates the force exerted by a single molecule on a wall and extends this to find the average force and pressure exerted by all molecules in the gas.

This paragraph delves into the kinetic theory of gases, explaining how the pressure of a gas is related to the kinetic energy of its molecules. It describes how the average kinetic energy of a molecule is proportional to the absolute temperature of the gas. The paragraph also discusses the difference between the microscopic view of temperature as molecular agitation and the macroscopic measurement of temperature. It highlights the concept that at absolute zero, all molecular motion ceases, which is why it is considered the lowest possible temperature.

The paragraph discusses the distribution of molecular velocities in a gas and how it relates to temperature. It explains that temperature does not correspond to a unique velocity but to a range of velocities described by the Maxwell-Boltzmann distribution. This distribution has a most probable value and is skewed, forced to vanish at the origin and at infinity. The paragraph also touches on the concept of specific heat for gases and how it differs from that of solids, emphasizing that specific heat for gases depends on whether the volume is allowed to change or not.

This paragraph connects the discussion on temperature and radiation to the cosmic background radiation predicted by the Big Bang theory. It explains how the current temperature of the universe, around 3 degrees Kelvin, is determined by observing the radiation from all directions in space, which fits the Black Body Radiation curve. This radiation is evidence of the Big Bang and provides insights into the universe's history and its current state of expansion or acceleration. The paragraph highlights the importance of understanding the distribution of energies at each frequency in relation to temperature, both for gases and radiation.

The paragraph introduces the First Law of Thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. It explains how internal energy can be changed by either doing work on the gas or adding heat. The paragraph also discusses the concept of a quasi-static process, which is a process that keeps the system in equilibrium at all times during a change in state. It emphasizes the importance of understanding the difference between a state and a process, and how the First Law of Thermodynamics applies to these processes.

This paragraph introduces the PV (pressure-volume) diagram as a tool for representing the states of a gas and how internal energy is related to these states. It defines internal energy as the kinetic energy of the gas molecules and provides the formula for calculating it in terms of pressure and volume (U = 3/2 PV). The paragraph explains that internal energy is a state function and depends only on temperature for an ideal gas. It also discusses the concept of specific heat for gases and how it differs from that of solids, emphasizing the need to consider moles rather than mass when dealing with gases.

The paragraph discusses the concept of specific heat for gases, differentiating between specific heat at constant volume (C_V) and specific heat at constant pressure (C_P). It explains how the specific heat of a gas depends on whether the volume is allowed to change or not during heating. The paragraph also highlights that the specific heat values provided are for monoatomic ideal gases, which do not account for additional energies associated with rotation or vibration in more complex molecules. It concludes with the ratio of C_P to C_V, known as the adiabatic index (γ), which is 5/3 for a monoatomic gas.