Question

please do all parts and show work

The goal of this question is to derive relationships of how the pressure and temperature (main part) depend on the height over the sea level h. To do so, we can consider thin slabs of air, vertically stacked. The temperature and pressure at sea level can be denoted T, and pe. For the numerical parts of this question, you can use a molar mass 29.0 g/mol, specific heat c, = 1.00 kJ/kg and cy = 0.718 kJ/kg, T.= 300 K and p. = 1 atm. a) Derive two expressions of dp/dh, one expressed using the density of air p., and one instead using the molar mass M and pressure p of the air. Use any appropriate approximations. b) Find an expression ph) of the pressure as a function of altitude (h) under the assumption that the temperature would not depend on h. What does this give for the pressure at the top of Mont Blanc (h = 4,810 m) and Mont Everest (h = 8,848 m)? c) Air is a fairly good insulator. It is hence reasonable to assume that the transport of air between different altitudes is adiabatic. Using this assumption, find an expression of the temperature as a function of height T(h). Evaluate the predicted temperatures at the heights specified in b). Tip: express Tin terms of p and differentiate this.

Answer 1

a) We know that

p=pgh

, the negative sign indicates the decrease of pressure with the increase in height.

Δp =-ρgh.

= -pg dh

Also from the gas equation:  

PV = RI

Mairp →p=Mair/V = RT

Substituting,

dp dh Mairpg RT

b) Solving we get,

p=Poe - Mair gh/RT

Using the values, the pressure at the top of Mt. Blanc

D=1 x -0.029x9.81x4810/(8.3145 x 300)

and that of Mt. Everest

D=1 x -0.029x9.81x8848/(8.3145 x 300)

c) For adiabatic process:

p1-T1 =const.
, where
7 = cp/cv = 1.4
, using the given values

(1-7) In p+yIn T = const.

, Substituting:

dp 7dT P 7-11

dT dh 7-1 Mairg R

T = T.-7-1 Mairgh R

Using the given values,

h T-300_1.4-10.029 x 9.81 1.4 8.3145

T = 300 - 9.78 x 10-3h

For Mt. Blanc

T = 300 - 9.78 x 10-3 X 4810 = 253K

For Mt. Everest

T = 300 - 9.78 x 10-3 x 8848 = 213K