The value of the ideal gas constant, R, is found to be as follows.
|=||53.34||ft·lbf·°R−1·lbm−1 (for air)|
The ideal gas law mathematically follows from a statistical mechanical treatment of primitive identical particles (point particles without internal structure) which do not interact, but exchange momentum (and hence kinetic energy) in elastic collisions.
Since it neglects both molecular size and intermolecular attractions, the ideal gas law is most accurate for monoatomic gases at high temperatures and low pressures. The neglect of molecular size becomes less important for larger volumes, i.e., for lower pressures. The relative importance of intermolecular attractions diminishes with increasing thermal kinetic energy i.e., with increasing temperatures. More sophisticated equations of state, such as the van der Waals equation, allow deviations from ideality caused by molecular size and intermolecular forces to be taken into account.
|Isobaric process|| || ||P2 = P1||V2 = V1 (V2/V1)||T2 = T1 (V2/V1)|
| || || ||P2 = P1||V2 = V1 (T2/T1)||T2 = T1 (T2/T1)|
|Isochoric process|| || ||P2 = P1 (P2/P1)||V2 = V1||T2 = T1 (P2/P1)|
| || || ||P2 = P1 (T2/T1)||V2 = V1||T2 = T1 (T2/T1)|
|Isothermal process|| || ||P2 = P1 (P2/P1)||V2 = V1 / (P2/P1)||T2 = T1|
| || || ||P2 = P1 / (V2/V1)||V2 = V1 (V2/V1)||T2 = T1|
| Isentropic process|
(Reversible adiabatic process)
| || ||P2 = P1 (P2/P1)||V2 = V1 (P2/P1) -1/||T2 = T1 (P2/P1)(-1)/|
| || || ||P2 = P1 (V2/V1) -||V2 = V1 (V2/V1)||T2 = T1 (V2/V1) 1-|
| || || ||P2 = P1 (T2/T1) /(-1)||V2 = V1 (T2/T1) 1/(1-)||T2 = T1 (T2/T1)|
Hence the ideal gas law
The ideal gas law can also be derived from first principles using the kinetic theory of gases, in which several simplifying assumptions are made, chief among which are that the molecules, or atoms, of the gas are point masses, possessing mass but no significant volume, and undergo only elastic collisions with each other and the sides of the container in which both linear momentum and kinetic energy are conserved.
Let q = (qx, qy, qz) and p = (px, py, pz) denote the position vector and momentum vector of a particle of an ideal gas,respectively, and let F denote the net force on that particle, then
By Newton's third law and the ideal gas assumption, the net force on the system is the force applied by the walls of their container, and this force is given by the pressure P of the gas. Hence
where dS is the infinitesimal area element along the walls of the container. Since the divergence of the position vector q is
the divergence theorem implies that
where dV is an infinitesimal volume within the container and V is the total volume of the container.
Putting these equalities together yields
which immediately implies the ideal gas law for N particles:
The readers are referred to the comprehensive article Configuration integral (statistical mechanics) where an alternative statistical mechanics derivation of the ideal-gas law, using the relationship between the Helmholtz free energy and the partition function, but without using the equipartition theorem, is provided.
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