InterviewSolution
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InterviewSolution
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Using the Maxwell distribution function, calculate the mean velocity projection `(v_x)` and the mean value of the modules of this projection `lt lt |v_x| gt gt` if the mass of each molecule is equal to `m` and the gas temperature is `T`. |
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Answer» `lt v_x gt = 0` bt symmetry `lt |v_x| gt = int_(- oo)^oo |v_x| e-(mv_x^2)/(2 kT) dv_x// int_0^oo (-mv_x^2)/(e 2kT) dv_x = int_0^oo v_x (-mv_x^2)/(e 2 kT) dv_x//int_0^oo (-mv^2)/(e 2 kT)dv_x` =`sqrt((2kT)/(m)) int_0^oo ue^(-u^2) du//int_0^oo e^(-u^2) du` =`sqrt((2kT)/(m)) int_0^oo (1)/(2) e^(-x) dx//int_0^oo e^(-x) (dx)/(2 sqrt(x))` =`sqrt((2 kT)/(m)) Gamma (1) //Gamma((1)/(2)) = sqrt((2 kT)/(m pi))`. |
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