Pressure and KMT Im going to quickly show the derivation pressure you arent

# Pressure and kmt im going to quickly show the

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Pressure and KMT I’m going to quickly show the derivation pressure, you aren’t responsible for the derivation – it’s shown to give you an idea of… … where this model comes from … how even simple models can give us deep insights into a system (the power of p-chem!!) Pressure is force per area Force is change in momentum per time For a particle (atom/molecule) traveling along x , the change in momentum in an elastic collision will be 𝐶ℎ???? 𝑖? ???????? ??? ????𝑖??? = 2 ? ? ? 8
Pressure and KMT, pt2 How many particles will hit the wall? If a particle is traveling at a speed of ? ? for some time ( Δ? ), then only particles within a distance of Δ? will collide with the wall. Δ? = ? ? Δ? If A is the total area of the wall, then only particles in a volume of 𝐴 ∙ Δ? can collide with the wall ?????? ?? ????𝑖???? ?ℎ?? ??? ℎ𝑖? ?ℎ? ???? = 𝐴 ? ? Δ? The density of particles in the volume is: 𝑃???𝑖??? ????𝑖?? = ? 𝑉 = ? ? 𝐴 𝑉 9
Pressure and KMT, pt3 How many particles will hit the wall (continued)? Velocity is assumed to be random, so only half of the particles have velocity of ? ? the other half travel at −? ? and won’t hit the wall ?????? ?? ????𝑖???? ℎ𝑖??𝑖?? ?ℎ? ???? ????? ? = ?????? ?? ????𝑖???? ?ℎ?? ??? ????? ?????ℎ ?? ℎ𝑖? ?ℎ? ???? × ????𝑖??? ????𝑖?? × 1 2 = ?∙? 𝐴 ∙𝐴∙𝑣 𝑥 ∙Δ? 2𝑉 Total change in momentum during time Δ t: ?ℎ???? 𝑖? ???????? = (?????? ?? ????𝑖????) × (???????? ?ℎ???? ??? ????𝑖???) = ? ∙ ? 𝐴 ∙ 𝐴 ∙ ? ? ∙ Δ? 2? 2?? ? 10
Pressure and KMT, pt4 (almost there) Force is change in momentum per time (note)N A ∙m = M, the molar mass Pressure is force per area And we expect to have a distribution of speeds, so we’ll use the average square velocity in the x -direction: <v x 2 > 11 ? Δ? = ? ∙ ? 𝐴 ∙ 𝐴 ∙ ? ? ∙ Δ? 2? 2?? ? = ? ∙ ? ∙ ? 𝐴 ∙ 𝐴 ∙ ? ? 2 ∙ Δ? ? ? = ? ∙ ? ∙ 𝐴 ∙ ? ? 2 ? ? = ? 𝐴 = ? ∙ ? ∙ 𝐴 ∙ ? ? 2 𝐴 ∙ ? = ? ∙ ? ∙ ? ? 2 ? ? = ? ∙ ? ∙ ? ? 2 ?
Pressure and KMT, pt5 (phew, and we did it!) We want total velocity, not just along the x -direction Which we get from a Pythagorean identity! No one direction is favored, so we expect to have the same average square velocity along all directions Lastly, as we will see in the next part (distribution of speeds) the square-root of the average square velocity is called the: root-mean-square velocity ( v rms ) 12 ? = ? ∙ ? ∙ ? ??? 2 3? ? 2 = ? ? 2 +? ? 2 +? ? 2 ? ? 2 = ? ? 2 = ? ? 2 ? 2 = 3 ? ? 2 ?? = ? ∙ ? ∙ ? ??? 2 3