Ideal Gas Law – A Supplemental Page

Ideal Gases: Independent molecules, not sticky, no V(r) but elastic collisions share energy. 3/2 kT = <1/2 mv²> = <KE> for one molecule. This relationship defines Temperature, T, and Boltzmann’s constant k = 1.38 x 10^-23 J/⁰C/molecule.

pV = NkT (microscopic view) where N = # of molecules.

A₀ = Avogadro’s Number of molecules = 6.02 x 10²³ molecules = 1 mole.  So, 6.02 x 10²³ protons = 1 gram of protons, i.e. the proton mass, m_p = 1.76 x 10^-24 grams, the reciprocal of A₀.  The ideal gas law may also be written pV = nRT  where n = # of moles and R = 8.31 J/⁰C/mol = 1.99 cal/⁰C/mol.

dQ = dU + dW   FIRST LAW OF THERMODYNAMICS

c_p n DT = 3/2nRDT + p  DV (at constant pressure)

The derivative of the Ideal Gas Law gives:  pdV + Vdp = nRdT

At constant pressure,  Vdp = 0, so  c_pnDT = 3/2nRDT + nRDT where c_p = 5/2 R = 5 cal/mol/⁰C.

At constant volume, pDV = 0, so  c_vnDT = 3/2nRDT where  c_v = 3/2 R = 3 cal/mol/⁰C.

DU for each process is given by DU = 3/2nRDT.

DU = 0 for isothermal processes.

DQ = 0 for adiabatic processes.

DQ = c_pnDT for constant pressure processes.

DQ = c_vnDT for constant volume processes.

DW = pDV and is zero for constant volume processes.

In adabatic processes the fall in pressure with increase in volume is faster than for isothermal processes because some of the internal energy is being used to do the work of expanding the gas.

For isothermal processes: pV = nRT = constant.

For adiabatic processes: pV^g = constant, where g = c_p/c_v = 5/3 = 1.667 for a monatomic gas, and 7/5 = 1.4 for a diatomic gas…

With the ideal gas law, the adiabatic processes also follow: TV^g-1 = constant.

The Otto Cycle is an approximation of the real ICE engine.Its efficiency is given by: N = 1 – (V₁/V₂)^1-g  where V₁/V₂ = the compression ratio, ~9 for an ICE engine.