Blundell Thermal Physics · Problem 6.6

Pressure

Physics — Statistical Mechanics · velocity-component-averages

Problem

The molecules in a gas travel with different velocities. A particular molecule will have velocity v\boldsymbol{v} and speed v=∣v∣v = |\boldsymbol{v}| and will move at an angle θ\theta to some chosen fixed axis. We have shown that the number of molecules in a gas with speeds between vv and v+dvv + \mathrm{d}v, and moving at angles between θ\theta and θ+dθ\theta + \mathrm{d}\theta to any chosen axis is given by 12 n f(v) dv sin⁡θ dθ,\frac{1}{2}\,n\,f(v)\,\mathrm{d}v\,\sin\theta\,\mathrm{d}\theta, where nn is the number of molecules per unit volume and f(v)f(v) is some function of vv only. [f(v)f(v) could be the Maxwellian distribution given above; however you should not assume this but rather calculate the general case.] Hence show by integration that: (a) ⟨u⟩=0\langle u \rangle = 0, (b) ⟨u2⟩=13⟨v2⟩\langle u^2 \rangle = \tfrac{1}{3}\langle v^2 \rangle, (c) ⟨∣u∣⟩=12⟨v⟩\langle |u| \rangle = \tfrac{1}{2}\langle v \rangle, where uu is any one Cartesian component of v\boldsymbol{v}, i.e., vxv_x, vyv_y, or vzv_z. [Hint: You can take uu as the zz-component of v\boldsymbol{v} without loss of generality. Why? Then express uu in terms of vv and θ\theta and average over vv and θ\theta. You can use expressions such as ⟨v⟩=∫0∞v f(v) dv∫0∞f(v) dv\langle v \rangle = \frac{\displaystyle\int_0^\infty v\,f(v)\,\mathrm{d}v}{\displaystyle\int_0^\infty f(v)\,\mathrm{d}v} and similarly for ⟨v2⟩\langle v^2 \rangle. Make sure you understand why.]

Related theory

Browse the problem library