a relation known as the barometric formula is usefulestimating the change in atm

Question

a relation known as the barometric formula is usefulestimating the change in atmospheric pressure with altitude.a)Starting with the knowledge that atmopheric pressure decreaseswith altitude, we have
dP= -pgdH where p is the density of air, g is the acceleration dueto gravity (9.81 m/s^2) and Pand h are the pressure and height,respectivley. Assuming ideal gas behavior and constant temperature,show that the pressure, P at the hight h is related to the pressureat sea level P0 (h=0) by P=P0e^(-gMh/RT) *Hint for an ideal gas,density=PM/RT, M=molar mass* b) calculate the atmospheric pressure at a height 5 km,assuming the temperature is constant at 5.0 degrees C given thatthe average molar mass of air is 29.0 g/mol. a relation known as the barometric formula is usefulestimating the change in atmospheric pressure with altitude.a)Starting with the knowledge that atmopheric pressure decreaseswith altitude, we have
dP= -pgdH where p is the density of air, g is the acceleration dueto gravity (9.81 m/s^2) and Pand h are the pressure and height,respectivley. Assuming ideal gas behavior and constant temperature,show that the pressure, P at the hight h is related to the pressureat sea level P0 (h=0) by P=P0e^(-gMh/RT) *Hint for an ideal gas,density=PM/RT, M=molar mass* b) calculate the atmospheric pressure at a height 5 km,assuming the temperature is constant at 5.0 degrees C given thatthe average molar mass of air is 29.0 g/mol.

Explanation / Answer

   We Know that :      As the distance of the altitudeincreases the pressure decreases with respect to their surface-area .          dP = - g dH with respect increase in theheight of the altitude            Assuming the gas to be an ideal behaviour and temperature isconstant.         dP = - PM / RT g dh         For an ideal behaviourof gas we can consider d = PM / RT         In order to obtainpressure at any point Integrate both sides we get :                  dP /  P   = - gM / RT   dh         lnP  = - gM / RT h + C                    Inorder to obtain the constant of integration We assume thecondition :              P = P0   ;    h= 0           C = ln P0          ln P / P0   = - gMh /RT          ln P / P0   = - gMh /RT                 P = P0   e-gMh / RT             hence the proof of the above formula derivation to obtain pressureat any point.     b )            P = P0 e-gMh / RT

                = 1 atm x e- 9.81 m / s2 * 29.0 g / mol * 5 * 1000m / ( 0.0821 atm-L / mol-K * 278 K )                      = ------- atm.

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