Question

(b) Deriving from basic and first principles of the kinetic theory of gases, prove that the root mean square speed of ideal gas molecules is given by [3RTIMjo 5, where M is the molar mass of the gas.

Answer 1

RMS SPEED USING KINETIC THEORY OF GAS LET US CONSIDER GAS ENCLOSED IN A CUBE TOTAL NUMBER OF MOLECULES = N MASS OF ONE MOLECULE = m TOTAL MASS OF GAS M� = m N ALONG X-AXIS N/3 MOLECULE INITIAL MOLECULE OF A MOLECULE P_1 = m v_1 MOMENTUM AFTER COLLISION P_2 = -m v_2 delta p = -m v_2 - m v_1 = -2 m v_1 v_1 = v_2 (ELASTIC COLLISION) TIME INTERVAL B/W TWO SUCCESSIVE COLLISION delta t = 2 l/v_1 FORCE EXERTED ON MOLECULE BY WALL = delta p/delta t = -m v^2_1/l FORCE EXERTED ON WALL BY MOLECULE F_Net = m v^2_1/l + m v^2_2/l + --- m v^2_N/l PRESSURE P = F/l^2 = m/l^3(v^2_1 + v^2_2 + ellipsis + v^2_N/3) RMS SPEED V_Rms = squareroot v^2_1 + v^2_2 + ellipsis + v^2_N/3/N/3 1/3 N V^2_RMS = P l^3/m l^3 = v VOLUME v^2_RMS = 3 p v/N m P V = n R T