Question

Relation of pressure and energy density. (a) Show that the average pres- sure in a system in thermal contact with a heat reservoir is given by {(res/CV)n expl – Es/T) (77) 2 where the sum is over all states of the system. (b) Show for a gas of free particles that (**). - 30. = WIN (78) CVN as a result of the boundary conditions of the problem. The result holds equally whether e, refers to a state of N noninteracting particles or to an orbital. (c) Show that for a gas of free nonrelativistic particles p = 2U/3V, (79) where U is the thermal average energy of the system. This result is not limited to the classical regime; it holds equally for fermion and boson particles, as long as they are nonrelativistic.

Answer 1

The pressure of an ideal gas is related to its energy density through the ideal gas law, PV = NKBT, where P represents pressure and T represents temperature. The average kinetic energy of gas molecules, which is proportional to temperature, directly relates to the pressure within the gas. For a gas of free nonrelativistic particles, pressure p equates to two-thirds the energy density (p = 2U/3V).

Step-by-step explanation:

Relation Between Pressure and Energy Density in an Ideal Gas

The pressure of an ideal gas and its energy density are related through the ideal gas law, which can be expressed as PV = NKBT. Here, P represents the macroscopic pressure, V is the volume, N is the number of molecules, and T is the temperature. Pressure is a measure of force per unit area, and in the context of an ideal gas, it is related to the kinetic energy of the gas molecules.

According to statistical mechanics, the average energy, represented as E = 1/2mv², where m is the mass of a molecule and v² is the average of the molecular speed squared, is directly proportional to the temperature (E = 3/2kT for each degree of freedom). This translational kinetic energy is an expression of the thermal energy of the molecules. Thus, the temperature of a gas is a measure of its average kinetic energy, and through the ideal gas law, we can relate the energy of a gas to both its pressure and volume.

In particular, for a gas of free nonrelativistic particles, the pressure is related to the thermal average energy U by the equation p = 2U/3V. This indicates that the pressure is two-thirds of the energy density, with U being the sum of the kinetic energies of all particles and V the volume occupied by the gas.