I recently did an experiment where I measured pressure at a few heights. The question I cam having trouble with is how to find the density of the air between the top and bottom. The difference in pressure from the top to bottom is 52.6 torr (692.5 to 745.1). The height of the stairwell, bottom to top measurement, was 17.094 meters. I am trying to find the ddensity in kg/m^3.

Question

I recently did an experiment where I measured pressure at a few heights. The question I cam having trouble with is how to find the density of the air between the top and bottom. The difference in pressure from the top to bottom is 52.6 torr (692.5 to 745.1). The height of the stairwell, bottom to top measurement, was 17.094 meters. I am trying to find the ddensity in kg/m^3.

Answer 1

There must have been something very wrong with your top-of-the-stairs measurement. Air pressure does not drop that much when increasing altitude by only 17 meters. That's only about 5 floors of a building. Atmspheric pressure decreases by a factor of about 0.87 (a 13% drop) per 1000 m altitude increase. Your measured decrease of about 7% sould have required, roughly, a 500 meter elevation gain.

The pressure change when the altitude changes by h equals the product of h, the (average) air density (rho), and the acceleration of gravity, g.
Delta P = (rho)*g*h
That pressure change will be in Newtons/sq. meter (also called Pascals). There are 760 torr per 1.013*10^5 Pascals, so
1 torr = 133.3 Pascals

You can use that formula to calculate the air density, if your pressure measurements are accurate.

Answer 2

To find the density of the air between the top and bottom, we can use the relationship between pressure, density, and height in a fluid. This relationship is given by the hydrostatic equation:

ΔP = ρ * g * Δh

Where:
ΔP is the change in pressure (745.1 - 692.5 = 52.6 torr),
ρ is the density of the fluid (air),
g is the acceleration due to gravity (approximately 9.8 m/s^2),
Δh is the change in height (17.094 meters).

We can rearrange the equation to solve for density (ρ):

ρ = ΔP / (g * Δh)

Now, let's substitute the values into the equation:

ρ = 52.6 torr / (9.8 m/s^2 * 17.094 m)

Before continuing, it's important to note that the unit for pressure needs to be converted to the SI unit of pascals (Pa) to ensure consistency in units.

1 torr = 133.322 Pa

Converting:

ρ = (52.6 torr * 133.322 Pa/torr) / (9.8 m/s^2 * 17.094 m)

Calculating:

ρ = 89.01 Pa / 168.1152 m^2

ρ ≈ 0.529 kg/m^3

Therefore, the approximate density of the air between the top and bottom is 0.529 kg/m^3.