Equation 2.12 provides the relationship between pressure and elevation in the atmosphere for those regions in which the temperature varies linearly with elevation. Derive this equation and verify the value of the pressure given in Table C. 2 in Appendix C for an elevation of \(5 \mathrm{~km}\).

Table C. 2

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Properties of the U.S. Standard Atmosphere (SI Units)* Acceleration Density, Altitude (m) Temperature of Gravity, (C) g (m/s) Pressure, p [N/m(abs)] Dynamic Viscosity, (kg/m) (N- s/m) -1,000 21.50 9.810 1.139 E+5 0 15.00 9.807 1.013 E +5 1.225 1.347 E + 0 E+0 1.821 E-5 1.789 E-5 1,000 8.50 9.804 8.988 E +4 1.112 E +0 1.758 E-5 2,000 2.00 9.801 7.950 E+4 1.007 E+0 1.726 E-5 3,000 -4.49 9.797 7.012 E + 4 9.093 E -1 1.694 E-5 4,000 -10.98 9.794 6.166 E +4 5,000 -17.47 9.791 5.405 E +4 6,000 -23.96 9.788 4.722 E +4 7,000 -30.45 9.785 4.111 E +4 8.194 E 7.364 E -1 6.601 E 1 5.900 E -1 - 1 1.661 E-5 1.628 E-5 1.595 E-5 1.561 E-5 8,000 -36.94 9.782 3.565 E+4 5.258 E 1 1.527 E-5 9,000 -43.42 9.779 3.080 E +4 4.671 E 1 1.493 E-5 10,000 -49.90 9.776 2.650 E +4 4.135 E 1 1.458 E-5 15,000 -56.50 9.761 1.211 E+4 1.948 E - 1 1.422 E-5 20,000 -56.50 9.745 5.529 E + 3 8.891 E - 2 1.422 E-5 25,000 -51.60 9.730 2.549 E + 3 4.008 E 2 1.448 E-5 30,000 -46.64 9.715 1.197 E3 1.841 E 2 1.475 E-5 40,000 -22.80 9.684 2.871 E +2 3.996 E 3 1.601 E 5 50,000 -2.50 9.654 7.978 E1 1.027 E 3 1.704 E-5 60,000 -26.13 9.624 2.196 E + 1 3.097 E - 4 1.584 E-5 70,000 -53.57 9.594 5.221 E+0 8.283 E - 5 1.438 E-5 80,000 -74.51 9.564 1.052 E + 0 1.846 E - 5 1.321 E-5