1. Let the variable z be the height above sea level. The barometric equation exp

Question

1. Let the variable z be the height above sea level. The barometric equation expresses the variation of air pressure P with altitude according to the relation dP mgP 4.81053 × 10-26 kg is the average mass of an air molecule, g 9.80665 m. s-2 is the × 10-23 J·K-1 is the Boltzmann where m constant. (a) According to the International Standard Atmosphere (ISA) model, the average temperature of the troposphere (that is, the bottommost 11 km of atmosphere) drops from its sea-level value of To 288.15 K at a steady rate of 6.5 K-km with increasing altitude. Use this information to determine the function T (z) for 0 3 z 3 11000 m. (b) The pressure at sea level is Po 1.01325 x 10s Pa according to the ISA model. Solve the harometric equation to get the pressure as a function of altitude.

Explanation / Answer

1. let variabel z be height above ground

then

dP/dz = -mgP/kT

m = 4.8105*10^-26 kg

k = 1.38*10^-23 J/K

g = 9.81 m/s/s

a. T(z = 0) = To = 288.15 K

dT/dz = -lambda = -6.5 K/m

T = -lambda*z + To

hence

b.

dP/P= mg*dz/k*lambda(z - To/lambda)

integrating from z = 0, P = Po to z = z, P = P

ln(P/Po) = mg*ln(1 - z*lambda/To)/k*lambda

P = Po*exp(mg*ln(1 - z*lambda/To)/k*lambda)

Po = 1.01325*10^5 Pa

P = 1.01325*10^5*exp(0.0052567474*ln(1 - 0.02255769564z)) Pa

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