User talk:Paul Wormer/scratchbook: Difference between revisions

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A  '''barometric formula''' gives the pressure in a planetary atmosphere—usually the [[Earth's atmosphere|Earth's]]—as a function of height. Since the [[temperature]] and [[composition]] of the atmosphere are  complicated functions of height, and because gravitation is  an inverse function of distance, a  completely general  barometric formula is very intricate.


However, two simplified formulas are in common use. The first is based on the assumption that temperature, gravitation,  and composition are constant throughout the atmosphere. Further it is used that the atmosphere is an [[ideal gas law|ideal gas]]. The equation thus obtained is a form of a [[Boltzmann distribution]],
:<math>
p(z) = p(z_0) \; \exp\left[ - \frac{mg}{RT} (z-z_0)\right].
</math>
Here ''p''(''z'')  is the pressure at height ''z''; ''z''<sub>0</sub> is a suitable reference height, often the surface of the planet and usually referred to as "base"; ''m'' is the  mass of one mole of the atmospheric gas (note that the equation holds for a pure gas, if the atmosphere contains more than one compound a weighted-average mass is used); ''g'' is the constant [[gravitational acceleration]]; ''R'' is the [[molar gas constant]]; ''T'' is the absolute [[temperature]] (assumed to be independent of ''z'').
The second barometric formula is based on the same assumptions as the first, the only generalization being that the temperature is taken to be linear in  height with a slope ''L'', the [[lapse rate]].
The equation is
:<math>
p(z) = p(z_0) \; \left[ \frac{T}{T + L (z-z_0)}\right]^{mg/(LR)}.
</math>
==Proof==
The two equations follow by integration of a simple differential equation that holds for the pressure of an ideal gas in a constant gravitational field.
===Differential equation===
{{Image|Pressure on pilbox.png|right|250px|A pillbox  of height &Delta;z and cross section ''O'' filled with air. The pillbox,  with upward pressure ''p'' at its bottom, is part of a cylinder filled with air.  Total mass ''M'' of air in pillbox is pulled downward by gravitational acceleration ''g'. }}
In the figure we see a stationary pillbox at arbitrary position ''z'' in a cylinder filled with an atmospheric gas that, for obvious reason,  we  call "air". The pillbox is pushed upward by a [[pressure]] force ''pO'', and pulled down by gravitation that acts with the force '''G''' equal to the total mass ''M'' of air in the pillbox times the [[gravitational acceleration]] ''g''. Also the air pressure (''p''+ &Delta;''p'')''O'' at the top acts downward. Forces upward are positive, downward negative, and since the total force is zero (no motion of air), we have
:<math>
pO -Mg - (p+\Delta p)O = 0 \;\Longrightarrow \; \Delta p = - \frac{Mg}{O}.
</math>
The total mass in the pillbox is the molar density &rho; times the molar mass ''m'' times the volume of the pillbox &Delta;''z''·''O''
:<math>
M = \rho\, m\, \Delta z O .\;
</math>
Hence
:<math>
\Delta p = -  \rho\, m\, g\, \Delta z .\,
</math>
Noting that the molar density is the number of moles per volume, we get for an [[ideal gas law|ideal gas]],
:<math>
p = \rho R T \;\Longrightarrow\; \rho = \frac{p}{RT}.
</math>
Assuming that the gas in the pillbox is ideal and of absolute [[temperature]] ''T'', we get
the equation
:<math>
\Delta p = - p \frac{mg}{RT} \Delta z \;\Longrightarrow\;
\frac{1}{p} \frac{\Delta p}{\Delta z} = -  \frac{mg}{RT} .
</math>
Taking  the limit of infinitesimally small  &Delta;  and
remembering that the derivative of the natural logarithm is given by
:<math>
\frac{1}{p(z)} \frac{dp(z)}{dz} = \frac{d\ln p(z)}{dz},
</math>
we arrive at the the differential equation
:<math>
\frac{d\ln\,p(z)}{dz} = - \frac{mg}{RT} . \qquad\qquad\qquad\qquad\qquad\qquad(1)
</math>
===Constant temperature===
The master equation (1) can be solved by integrating the left hand side from ''p''(''z''<sub>0</sub>) &equiv; ''p''<sub>0</sub> to
''p''(''z'') and the right hand side from ''z''<sub>0</sub> to ''z''. Hence
:<math>
\ln\frac{p}{p_0} = - \frac{mg}{RT} (z-z_0), \qquad\qquad\qquad\qquad\qquad(2)
</math>
where we recall that
:<math>
\ln p - \ln p_0 = \ln\frac{p}{p_0}.
</math>
Taking the exponent (the inverse function of the natural logarithm) of both sides of equation (2)
we get
:<math>
\frac{p}{p_0} = \frac{\exp\left[-  \frac{mg}{RT} z\right]}{\exp\left[-  \frac{mg}{RT} z_0\right]}
\;\Longleftrightarrow\;
p = p_0 \, \exp\left[ - \frac{mg}{RT}(z-z_0) \right].
</math>
This  distribution is known in [[statistical thermodynamics]] as a [[Boltzmann distribution]].
===Temperature linear in height===
We assumed in the previous subsection that ''T'' is constant throughout the cylinder, which makes the equation useful for cases where there is no temperature gradient, i.e., cases where the temperature at the top of the cylinder is equal to the temperature at the bottom.  The master equation (1) can be modified for the case of a constant temperature gradient (constant [[lapse rate]]). That is, we  make in equation (1) the substitution
:<math>
T \rightarrow T + L (z-z_0).
</math>
The modified equation is:
:<math>
\int_{p_0}^{p} d\ln\,p'  = - \frac{mg}{R} \int_{z_0}^z \frac{1}{T + L (z'-z_0)} dz'  .
</math>
Substitute
:<math>
y = T +L(z'-z_0)\; \Longrightarrow \; dz'= \frac{1}{L} dy .
</math>
The integral on the right hand side becomes
:<math>
- \frac{mg}{L R} \int^{T+L(z-z_0)}_T \frac{dy}{y}  =
- \frac{mg}{L R} \ln \frac{T+L(z-z_0)}{T} =
\ln\;\left[ \frac{T}{T+L(z-z_0)} \right]^{\frac{mg}{L R}}.
</math>
We already saw that the integral on the left hand side is the logarithm of ''p''/''p''<sub>0</sub>. hence
:<math>
\frac{p}{p_0} = \left[ \frac{T}{T+L(z-z_0)} \right]^{\frac{mg}{L R}}.
</math>

Revision as of 04:03, 27 August 2009

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