# Using Pressure and Temperature differences to find height

• M_Abubakr
In summary, the conversation discusses finding the height of a mountain given the temperature and pressure readings at the top and bottom of the mountain. The attempt at a solution involves using the temperature lapse rate, linear equations, and the ideal gas law. However, the assumption of constant air density leads to an incorrect answer. Instead, integrating the change in pressure with height and matching the pressures at the foot and top of the mountain is necessary for an accurate calculation.
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.

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

## 1. How does pressure and temperature affect the calculation of height?

Pressure and temperature are two key factors that affect the density of air. As altitude increases, the air becomes less dense, leading to a decrease in pressure and temperature. This decrease in pressure and temperature can be used to calculate the height of a location.

## 2. What is the formula for using pressure and temperature differences to find height?

The formula for using pressure and temperature differences to find height is:
h = (RT/gM) * ln(P1/P2)
Where:
h = height in meters
R = gas constant (8.314 J/mol*K)
T = temperature (in Kelvin)
g = gravitational acceleration (9.8 m/s²)
M = molar mass of air (0.02896 kg/mol)
P1 = pressure at known height
P2 = pressure at unknown height

## 3. Is this method accurate for calculating height?

Yes, using pressure and temperature differences to find height is a reliable method that has been used for many years. However, it is important to note that this method assumes a standard atmosphere and may not be as accurate in regions with extreme weather conditions or at higher altitudes.

## 4. What instruments are used to measure pressure and temperature for this calculation?

The most commonly used instruments for measuring pressure and temperature are barometers and thermometers. Barometers measure atmospheric pressure and can be either mercury or aneroid type. Thermometers measure air temperature and can be either liquid-in-glass or digital.

## 5. Can this method be used to calculate height on other planets or moons?

Yes, this method can be used to calculate height on other planets or moons as long as the necessary data for pressure and temperature are available. However, the formula may need to be adjusted based on the specific atmospheric conditions of the planet or moon being measured.

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