<P> 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 . </P> <P> Let q = (q, q, q) and p = (p, p, p) denote the position vector and momentum vector of a particle of an ideal gas, respectively . Let F denote the net force on that particle . Then the time - averaged potential energy of the particle is: </P> <Dl> <Dd> ⟨ q ⋅ F ⟩ = ⟨ q x d p x d t ⟩ + ⟨ q y d p y d t ⟩ + ⟨ q z d p z d t ⟩ = − ⟨ q x ∂ H ∂ q x ⟩ − ⟨ q y ∂ H ∂ q y ⟩ − ⟨ q z ∂ H ∂ q z ⟩ = − 3 k B T, (\ displaystyle (\ begin (aligned) \ langle \ mathbf (q) \ cdot \ mathbf (F) \ rangle & = (\ Bigl \ langle) q_ (x) (\ frac (dp_ (x)) (dt)) (\ Bigr \ rangle) + (\ Bigl \ langle) q_ (y) (\ frac (dp_ (y)) (dt)) (\ Bigr \ rangle) + (\ Bigl \ langle) q_ (z) (\ frac (dp_ (z)) (dt)) (\ Bigr \ rangle) \ \ & = - (\ Bigl \ langle) q_ (x) (\ frac (\ partial H) (\ partial q_ (x))) (\ Bigr \ rangle) - (\ Bigl \ langle) q_ (y) (\ frac (\ partial H) (\ partial q_ (y))) (\ Bigr \ rangle) - (\ Bigl \ langle) q_ (z) (\ frac (\ partial H) (\ partial q_ (z))) (\ Bigr \ rangle) = - 3k_ (B) T, \ end (aligned))) </Dd> </Dl> <Dd> ⟨ q ⋅ F ⟩ = ⟨ q x d p x d t ⟩ + ⟨ q y d p y d t ⟩ + ⟨ q z d p z d t ⟩ = − ⟨ q x ∂ H ∂ q x ⟩ − ⟨ q y ∂ H ∂ q y ⟩ − ⟨ q z ∂ H ∂ q z ⟩ = − 3 k B T, (\ displaystyle (\ begin (aligned) \ langle \ mathbf (q) \ cdot \ mathbf (F) \ rangle & = (\ Bigl \ langle) q_ (x) (\ frac (dp_ (x)) (dt)) (\ Bigr \ rangle) + (\ Bigl \ langle) q_ (y) (\ frac (dp_ (y)) (dt)) (\ Bigr \ rangle) + (\ Bigl \ langle) q_ (z) (\ frac (dp_ (z)) (dt)) (\ Bigr \ rangle) \ \ & = - (\ Bigl \ langle) q_ (x) (\ frac (\ partial H) (\ partial q_ (x))) (\ Bigr \ rangle) - (\ Bigl \ langle) q_ (y) (\ frac (\ partial H) (\ partial q_ (y))) (\ Bigr \ rangle) - (\ Bigl \ langle) q_ (z) (\ frac (\ partial H) (\ partial q_ (z))) (\ Bigr \ rangle) = - 3k_ (B) T, \ end (aligned))) </Dd>

Si units for pressure in ideal gas law