The Maxwell-Boltzmann speed distribution is the probability distribution that describes the speeds of the particles of ideal gases. The Maxwell-Boltzmann speed distribution is valid for both un-mixed particles (one type of particle) and mixed particles (two types of particles). For mixed particles, both types of particles follow the Maxwell-Boltzmann speed distribution. Also, the most probable speed is inversely proportional to the square root of the mass.
This paper proves the Maxwell-Boltzmann speed distribution and the speed ratio of mixed particles using computer-generated data based on Newton’s law of motion. To achieve this, this paper derives the probability density function ψ^ab(u_a;v_a,v_b) of the speed u_a of the particle with mass M_a after the collision of two particles with mass M_a in speed v_a and mass M_b in speed v_b. The function ψ^ab(u_a;v_a,v_b) is obtained through a unique procedure that considers (1) the randomness of the relative direction before a collision by an angle α. (2) the randomness of the direction after the collision by another independent angle.
The function ψ^ab(u_a;v_a,v_b) is used in the equation below for the numerical iterations to get new distributions P_new^a(u_a) from old distributions P_old^a(v_a), and repeat with P_old^a(v_a)=P_new^a(v_a), where n_a is the fraction of particles with mass M_a.
P_new^1(u_1)=n_1 ∫_0^∞ ∫_0^∞ ψ^11(u_1;v_1,v’_1) P_old^1(v_1) P_old^1(v’_1) dv_1 dv’_1
+n_2 ∫_0^∞ ∫_0^∞ ψ^12(u_1;v_1,v_2) P_old^1(v_1) P_old^2(v_2) dv_1 dv_2
P_new^2(u_2)=n_1 ∫_0^∞ ∫_0^∞ ψ^21(u_2;v_2,v_1) P_old^2(v_2) P_old^1(v_1) dv_2 dv_1
+n_2 ∫_0^∞ ∫_0^∞ ψ^22(u_2;v_2,v’_2) P_old^2(v_2) P_old^2(v’_2) dv_2 dv’_2
The final distributions converge to the Maxwell-Boltzmann speed distributions. Moreover, the square of the root-mean-square speed from the final distribution is inversely proportional to the particle masses as predicted by Avogadro’s law.
Mean square solutions of random linear models and computation of their probability density function
