In this problem, we will derive a law relating the pressure gradient m the atmos

Question

In this problem, we will derive a law relating the pressure gradient m the atmosphere to temperature, gravity and pressure. This can be used to calculate the atmospheric pressure as a function of height under the assumptions of constant temperature and composition. A pressure gradient exists in the atmosphere to balance the gravitational pull. Look at a thin slab of air and write down an expression for dp/dz in terms of air density. This is the equation of hydrostatic equilibrium. Use the ideal gas law and the average molecular mass m to find the density in terms of thermodynamic variables. Substitute into the result of part (a) to get dP/dz = -mg P / kT

Explanation / Answer

P(z) = P(z + dz) + *dz*g -g = dP/dz   (from previous answer) b) = mass/volume = (molecules/volume)*molecular mass =(N/V)*m PV = NkT       =>   N/V= (P/kT) = (P/kT)*m Sub into -g = dP/dz dP/dz = -(mg/kT)*P c) dP/P = -(mg/kT)*dz lnP - lnP0   = -(mg/kT)*(z - 0) = ln(P/P0) P/P0 = exp(-mg*z/kT) P = P0*exp(-mg*z/kT)

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