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In an ideal mixture each gas exerts pressure in proportion to its share of the molecules, so the partial pressure is the mole fraction times the total pressure. One mole of a component in four moles of mixture at 2 atm contributes 0.5 atm. Because mole fraction equals volume fraction for ideal gases, the same number describes the composition by volume.
Partial pressure
p(i) = x(i) x P(total)
Mole fraction
x(i) = moles of component / total moles
For ideal gases, yes. That is why atmospheric oxygen is quoted as 21% by volume and also has a mole fraction of 0.21.
Not directly - use Henry's law for solubility. But the partial pressure computed here is the driving pressure that Henry's law needs.