At the top of a mountain the temperature is -5 degree C and a mercury barometer reads 566 mm, whereas the reading at the foot of the mountain is 749 mm.
Assuming a temperature lapse rate of 0.0065 K/m and R = 287 J/kg K, calculate the height of the mountain.
The Attempt at a Solution
The lapse rate tells me how much temperature increases or decreases with height. So I can say that Temperature is a function of height T(d). And I'm assuming it to be linear function so I used the equation of a line which is y=mx+c.
and c is the y intercept which is the temperature when height is 0 so the equation will be
I've taken slope as negative because temperature decreases with change in height which is obvious.
The pressure at the peak will be (from the mercury barometer reading)
The pressure at the foot of the mountain is: (also from the mercury barometer reading)
From the formula PV=mRT
we get P=ρRT
I assumed that the density of air will remain constant with change in height.
so what I did was I made rho subject.
so at peak
and at the foot
Cancelling R on both sides will give us
making T2 subject will give us the temperature at the foot of the mountain.
Which is the temperature at the foot of the mountain. Its also the y intercept c of the linear function T(d)= -0.0065d+c because it tells us the temperature when height is zero.
so the equation we'll end up with will be
To find the height of the peak from the foot of the mountain we'll put T(d) = 273-5 = 268K
268 = -0.0065d+354.65
after making d subject and solving further we'll get d=13330.76923m
Can anyone tell me what I did wrong in my attempt to solve this question? I know this is not the correct answer.