Gas molecules move around inside a box with sides of length 1 μm. Each time a molecule rebounds from a wall, it pushes on the wall.
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全分子の向きを変えずに速さを2倍にすると、右の壁の圧力は何倍？
x方向の速度成分だけを2倍にすると、上の壁の圧力は何倍？
The equations explain the phenomenon
1The equation of motion is set upmdvdt=F……
2The mass is moved inside the derivatived(mv)dt=F……
3The impulse during the collision is calculatedΔ(mv)=∫F dt……
4The difference between the momenta before and after the collision is calculated｜Δ(mvx)｜=2mvx……
5Collisions with the same wall are countedNumber of collisions≒|vx| t/2L……
6The impulse per collision is multiplied by the number of collisionsTotal impulse≒2m|vx|×|vx| t / 2L……
7The factors of 2 cancel, and |vx| × |vx| becomes vx²Total impulse≒m vx² t / L……
8Dividing by time gives the average forceAverage force on the wall=m vx² / L……
9Dividing by the wall area gives the pressurep₁=m vx² / L³……
10L³ is replaced by the volume Vp₁=m vx² / V……
11The pressure contributions from N molecules are addedp=Nm ⟨vx²⟩ / V……
12The mean squares of the velocity components in the 3 directions are taken to be equalp=⅓(NV)m ⟨v²⟩……
Further study connects molecular motion and temperature
13The equation of state connects molecular motion to temperature½ m ⟨v²⟩=(32) k T……
What this chapter has shown
The magnitude of the impulse delivered to the right wall by a molecule in a single collision is 2m|vx|. Adding the impulses delivered by all molecules over a sufficiently long time and dividing by that time and the wall area gives the average pressure p=N m⟨vx²⟩/V.
4 timesThe speeds of all molecules have been doubled
Doubling only the x component of velocity makes the pressure on the left and right walls 4 times its original value, while the pressure on the top and bottom walls remains unchanged (a factor of 1).
In everyday lifeAt the same temperature and volume, increasing the number of molecules increases the number of collisions with the walls and raises the pressure. A tire becomes firmer when air is pumped into it because the pressure pushing outward from inside increases.
Detailed summary
Gas pressure is the total impulse delivered by molecules as they rebound from a wall, divided by the elapsed time to obtain the average force, and then by the wall area. Both the impulse from 1 collision and the collision rate are proportional to vx, so the pressure is proportional to the average of vx raised to the power 2. If molecular motion has no preferred direction, p=⅓(N/V)m⟨v²⟩. Comparing this result with the equation of state shows that temperature is proportional to the average translational kinetic energy of 1 molecule.