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Isotherms of an ideal gas for different temperatures. The curved lines are rectangular hyperbolae of the form y = a/x. They represent the relationship between pressure (on the vertical axis) and volume (on the horizontal axis) for an ideal gas at different temperatures: lines that are farther away from the origin (that is, lines that are nearer to the top right-hand corner of the diagram ...
v = velocity of atom/molecule, m = mass of each molecule (all molecules are identical in kinetic theory), γ(p) = Lorentz factor as function of momentum (see below) Ratio of thermal to rest mass-energy of each molecule: = /
P = pressure V = volume n = number of moles R = universal gas constant T = temperature. The ideal gas equation of state can be arranged to give: = / or = / The following partial derivatives are obtained from the above equation of state:
In physics, the thermal equation of state is a mathematical expression of pressure P, temperature T, and, volume V.The thermal equation of state for ideal gases is the ideal gas law, expressed as PV=nRT (where R is the gas constant and n the amount of substance), while the thermal equation of state for solids is expressed as:
For example, the ideal gas law in terms of the Boltzmann constant is: =, where N is the number of particles (molecules in this case), or to generalize to an inhomogeneous system the local form holds: =, where ρ N = N/V is the number density.
In thermodynamic terms, this is a consequence of the fact that the internal pressure of an ideal gas vanishes. Mayer's relation allows us to deduce the value of C V from the more easily measured (and more commonly tabulated) value of C P: =.
It is an intermediate mathematical model, useful as a pedagogical tool when teaching physics, chemistry, and engineering. In addition, its saturation curve has an analytic solution, which can depict the liquid metals (mercury and cesium) quantitatively, and describes most real fluids qualitatively. [ 25 ]
The first and second law of thermodynamics are the most fundamental equations of thermodynamics. They may be combined into what is known as fundamental thermodynamic relation which describes all of the changes of thermodynamic state functions of a system of uniform temperature and pressure.