Using Pressure and Temperature differences to find height

[M_Abubakr]

Homework Statement

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.

Homework Equations

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.
where
y=T(d)
m=0.0065K/m
x=d
and c is the y intercept which is the temperature when height is 0 so the equation will be

T(d)= -0.0065d+c

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)
P1=ρgh1
P1=13560x9.81x566x10^-3
P1= 75291.3576m

The pressure at the foot of the mountain is: (also from the mercury barometer reading)
P2=ρgh2
P2=13560x9.81x749x10^-3
P2= 99634.6764m

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.
ρ=P/RT
so at peak
ρ=P1/RT1
and at the foot
ρ=P2/RT2
ρ=ρ
P1/RT1=P2/RT2
Cancelling R on both sides will give us
P1/T1=P2/T2
making T2 subject will give us the temperature at the foot of the mountain.

T2=P2T1/P1
T2=(99634.67640)*(273-5)/(75291.3576)
T2=354.65
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

T(d)= -0.0065d+354.65
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.

 

[scottdave]

Perhaps this hint might help. https://www.engineeringtoolbox.com/mercury-d_1002.html

 

[Chestermiller]

If the temperature at height d is -5C, and the lapse rate is 0.0065 C/m, in terms of d, what is the temperature at the ground? In terms of d, what is the absolute temperature as a function of elevation z and height d?

Assuming the density of the air is constant was not a good idea. For arbitrary locations, $$\frac{dP}{dz}=-\rho g=\frac{P}{RT}g$$You need to integrate this between z = 0 and z = d, and match the pressures at the foot and at z = d.

 

[Dr Dr news]

Good suggestion with the z=0 being sea level and p and T being the sea level standard values.