The Kinetic Theory of ideal gases
- It is employed to express quantitatively the ideal gas model. Postulates:
1- A gas consists of molecules of mass (m) and diameter (d). 2- Molecules move continuously and randomly.
3- Molecules are treated as points, having no volume
4- Molecules collide with each other, changing direction and velocity. 5- Collisions are elastic (no loss of translational energy) no potential energy of interaction between them.
Derivation of the ideal gas equation from the kinetic theory
The model used for discussing the molecular basis of the physical properties of a perfect gas. The molecules move chaotically with a range of speeds and directions, both of which change when they collide with the walls or with other molecules.
- Consider the system in Fig. 1
- A particle of mass m collides with the wall with speed u parallel to the x-axis, the impact is mux, (its component of linear momentum) - It rebounds with speed -ux.
- Overall change of momentum = mux – (-mux) = 2mux.
- In interval (t, the number of collisions = number of particles able to reach the wall in (t. - All particles within a distance ux(t of the wall will strike it if they are traveling towards it. - If the wall of area A, all particles in volume Aux(t will reach the wall. - If the number of particles per unit volume is N, the number in the volume = NAux(t - On average, half the particles are moving to the right, hence, the average number of collisions = ½ NAux(t - Total momentum change = ½ NAux (t x 2mux = mNAu2x (t
- Rate of change of momentum = mNAu2x (By Newton’s second law it equals to the force = p/A) - Pressure = mNu2x
- But the detected pressure is the average one, so p= mN - [pic]
- Particles are moving randomly, so speeds in x, y, z direction are all equal, then c = (3)1/2, implying that = 1/3 c2 - Therefore, p= 1/3 Nmc2
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