Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 3.4-11
SOME FORMAL RELATIONSHIPS, AND SAMPLE SYSTEMS
Physics — Statistical Mechanics · partial-pressures-multicomponent-ideal-gas
Problem
Show that the pressure of a multicomponent simple ideal gas can be written as the sum of ``partial pressures'' , where . These ``partial pressures'' are purely formal quantities not subject to experimental observation. (From the mechanistic viewpoint of kinetic theory the partial pressure is the contribution to the total pressure that results from bombardment of the wall by molecules of species —a distinction that can be made only when the molecules are noninteracting, as in an ideal gas.)Intuitive explanation
The key insight is that each gas species in an ideal gas mixture behaves independently because molecules don't interact—so you can apply the ideal gas law to each component separately, then add the results. Start by writing the total pressure from the equation of state for a mixture, then factor out how many moles of each species contribute to that pressure. The "partial pressure" is just the pressure that species would exert if it alone occupied the entire volume at the same temperature—a useful bookkeeping tool that reveals how the total pressure decomposes into independent molecular contributions.
Intuitive explanation
The key insight is that each gas species in an ideal gas mixture behaves independently because molecules don't interact—so you can apply the ideal gas law to each component separately, then add the results. Start by writing the total pressure from the equation of state for a mixture, then factor out how many moles of each species contribute to that pressure. The "partial pressure" is just the pressure that species would exert if it alone occupied the entire volume at the same temperature—a useful bookkeeping tool that reveals how the total pressure decomposes into independent molecular contributions.