My name is Alexander FufaeV and here I write about:
How Does a Barometer Work?
Important Formula
$$\mathit{\Pi}(h) ~=~ \mathit{\Pi}_0 \, \mathrm{e}^{-h/H}$$
$$\mathit{\Pi}(h) ~=~ \mathit{\Pi}_0 \, \mathrm{e}^{-h/H}$$
$$\mathit{\Pi}_0 ~=~ \frac{ \mathit{\Pi} }{ \mathrm{e}^{-h/H} }$$
$$H ~=~ - \frac{h}{ \ln\left( \mathit{\Pi}/\mathit{\Pi}_0 \right) }$$
$$h ~=~ - H \, \ln\left( \frac{\mathit{\Pi}}{\mathit{\Pi}_0} \right)$$
What do the formula symbols mean?
Pressure
$$ \mathit{\Pi} $$ Unit $$ \mathrm{Pa} = \frac{ \mathrm{N} }{ \mathrm{m}^2 } $$
Air pressure (or the pressure of another fluid) at height \(h\) above the sea surface. According to this barometric formula, air pressure decreases exponentially with height.
Pressure at the bottom
$$ \mathit{\Pi}_0 $$ Unit $$ \mathrm{Pa} $$
Air pressure at sea level \(h = 0\).
Scale height
$$ H $$ Unit $$ \mathrm{m} $$
Scale height is a constant that depends, for example, on the temperature of the air and the gravitational acceleration. The scale height is approximately \(H=8\,\mathrm{km}\) on Earth.
Height
$$ h $$ Unit $$ \mathrm{m} $$
Height measured from the sea surface.
A simple barometer works by using the barometer formula, which relates the height \(h\) above the ground to the air pressure \( \mathit{\Pi} \) prevailing at that height:
$$ \begin{align} \mathit{\Pi}(h) ~=~ \mathit{\Pi}_0 \, \mathrm{e}^{-h/H} \end{align} $$
Here \( H ~=~ 8005 \, \mathrm{m} \) and the mean air pressure at sea level is \(\mathit{\Pi}_0 = 101 \, \mathrm{kPa} = 1 \, \mathrm{bar} \). The formula states that the air pressure decreases exponentially with increasing height \( h \).
So an barometer measures the air pressure \( \mathit{\Pi} \) and calculates from it the height at which this air pressure prevails:
$$ \begin{align} h ~=~ 8005 \, \mathrm{m} \,\cdot\, \ln\left(\frac{ \mathit{\Pi} }{ 101 000 \, \mathrm{Pa} }\right) \end{align} $$