Blundell Thermal Physics · Problem 6.9

Pressure

Physics — Statistical Mechanics · pressure-molecular-flux-alternative-derivation

Problem

This question provides an alternative derivation of the formula for pressure. Without loss of generality, let us consider molecules travelling towards a wall which lies in the xyxy plane. The momentum change of a molecule of mass mm and velocity v=(vx,vy,vz)\boldsymbol{v} = (v_x, v_y, v_z) bouncing off the wall will be 2mvx2mv_x. Explain why the pressure pp on the wall is given by p=∫0∞(2mvx) vx n g(vx) dvx,(6.29)p = \int_0^\infty (2mv_x)\,v_x\,n\,g(v_x)\,\mathrm{d}v_x, \tag{6.29} where g(vx)g(v_x) is the function given in eqn 5.2. Hence show that p=nkBTp = nk_\mathrm{B}T. Using the same approach show that Φ\Phi, the number of molecules hitting unit area of the wall per second is given by Φ=∫0∞vx n g(vx) dvx=nkBTπm=14n⟨v⟩.(6.30)\Phi = \int_0^\infty v_x\,n\,g(v_x)\,\mathrm{d}v_x = n\sqrt{\frac{k_\mathrm{B}T}{\pi m}} = \frac{1}{4}n\langle v \rangle. \tag{6.30} This result will be derived using a different method in the next chapter.

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