1. Idea Behind the Derivation
The goal is to connect the pressure of a gas with the motion of its molecules. If we assume gas molecules move freely and collide elastically with the container walls, each collision gives a tiny push. Adding up all these pushes gives the pressure.
The derivation shows how pressure depends on the number of molecules and the average of their squared speeds.
2. Assumptions Used
To derive the pressure formula, we assume:
- Molecules behave like point masses.
- Collisions with walls are perfectly elastic.
- Motion is random and equally likely in all directions.
- No intermolecular forces except during collisions.
2.1. Why These Assumptions Help
They allow us to treat molecular motion using simple mechanics without worrying about forces between molecules.
3. Momentum Change During Wall Collision
Consider one molecule of mass m moving with velocity cx in the x-direction. When it hits the wall and bounces back elastically, its velocity becomes -cx.
So the change in momentum is:
\( \Delta p = m(-c_x) - m(c_x) = -2mc_x \)
3.1. Force on the Wall
The molecule keeps hitting the same wall repeatedly. Time between two collisions with the same wall is:
\( \Delta t = \dfrac{2L}{c_x} \)
Thus, average force exerted by one molecule is:
\( F = \dfrac{\Delta p}{\Delta t} = \dfrac{2mc_x}{\dfrac{2L}{c_x}} = \dfrac{mc_x^2}{L} \)
4. Pressure Due to Many Molecules
If there are N molecules in the container, the total force is the sum of forces from all molecules. Pressure is force divided by area.
The resulting expression becomes:
\( P = \dfrac{1}{V} m (c_{x1}^2 + c_{x2}^2 + ... + c_{xN}^2) \)
4.1. Using Symmetry
Because motion is random in all directions:
\( \bar{c_x^2} = \bar{c_y^2} = \bar{c_z^2} = \dfrac{1}{3}\bar{c^2} \)
Thus:
\( P = \dfrac{1}{3} \rho \bar{c^2} \)
5. Final Pressure Relation
The derived formula connecting pressure to molecular motion is:
\( \boxed{ P = \dfrac{1}{3} \rho \bar{c^2} } \)
Where:
5.1. Meaning of the Result
Pressure increases if:
- molecules move faster (higher speed)
- molecules are heavier
- there are more molecules in a given volume