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searchaux" style="display:none">Variation in pressure as a function of elevation</div> <p><b>Vertical pressure variation</b> is the variation in <a href="/wiki/Pressure" title="Pressure">pressure</a> as a function of <a href="/wiki/Elevation" title="Elevation">elevation</a>. Depending on the <a href="/wiki/Fluid" title="Fluid">fluid</a> in question and the context being referred to, it may also vary significantly in dimensions perpendicular to elevation as well, and these variations have relevance in the context of <a href="/wiki/Pressure_gradient_force" class="mw-redirect" title="Pressure gradient force">pressure gradient force</a> and its effects. However, the vertical variation is especially significant, as it results from the pull of <a href="/wiki/Gravity" title="Gravity">gravity</a> on the fluid; namely, for the same given fluid, a decrease in elevation within it corresponds to a taller column of fluid weighing down on that point. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Basic_formula">Basic formula</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertical_pressure_variation&action=edit&section=1" title="Edit section: Basic formula"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A relatively simple version <sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> of the vertical fluid pressure variation is simply that the pressure difference between two elevations is the product of elevation change, gravity, and <a href="/wiki/Density" title="Density">density</a>. The equation is as follows: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dP}{dh}}=-\rho g,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>P</mi> </mrow> <mrow> <mi>d</mi> <mi>h</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>ρ</mi> <mi>g</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dP}{dh}}=-\rho g,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b3622c11dde02bac8964f72a11c8784169f63e9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.669ex; height:5.509ex;" alt="{\displaystyle {\frac {dP}{dh}}=-\rho g,}"></span> where </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">P</span> is pressure,</li> <li><span class="texhtml mvar" style="font-style:italic;">ρ</span> is density,</li> <li><span class="texhtml mvar" style="font-style:italic;">g</span> is <a href="/wiki/Gravity_of_Earth" title="Gravity of Earth">acceleration of gravity</a>, and</li> <li><span class="texhtml mvar" style="font-style:italic;">h</span> is height.</li></ul> <p>The delta symbol indicates a change in a given variable. Since <span class="texhtml mvar" style="font-style:italic;">g</span> is negative, an increase in height will correspond to a decrease in pressure, which fits with the previously mentioned reasoning about the weight of a column of fluid. </p><p>When density and gravity are approximately constant (that is, for relatively small changes in height), simply multiplying height difference, gravity, and density will yield a good approximation of pressure difference. If the pressure at one point in a liquid with uniform density ρ is known to be P<sub>0</sub>, then the pressure at another point is P<sub>1</sub>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}=P_{0}-\rho g(h_{1}-h_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>ρ</mi> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}=P_{0}-\rho g(h_{1}-h_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb075843559d4d300a19d0f41df54dcb1f5a6c0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.786ex; height:2.843ex;" alt="{\displaystyle P_{1}=P_{0}-\rho g(h_{1}-h_{0})}"></span></dd></dl> <p>where h<sub>1</sub> - h<sub>0</sub> is the vertical distance between the two points.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Where different fluids are layered on top of one another, the total pressure difference would be obtained by adding the two pressure differences; the first being from point 1 to the boundary, the second being from the boundary to point 2; which would just involve substituting the <span class="texhtml mvar" style="font-style:italic;">ρ</span> and <span class="texhtml">Δ<i>h</i></span> values for each fluid and taking the sum of the results. If the density of the fluid varies with height, mathematical <a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">integration</a> would be required. </p><p>Whether or not density and gravity can be reasonably approximated as constant depends on the level of <a href="/wiki/Accuracy_and_precision" title="Accuracy and precision">accuracy</a> needed, but also on the <a href="/wiki/Length_scale" title="Length scale">length scale</a> of height difference, as gravity and density also decrease with higher elevation. For density in particular, the fluid in question is also relevant; <a href="/wiki/Seawater" title="Seawater">seawater</a>, for example, is considered an <a href="/wiki/Incompressible_fluid" class="mw-redirect" title="Incompressible fluid">incompressible fluid</a>; its density can vary with height, but much less significantly than that of air. Thus water's density can be more reasonably approximated as constant than that of air, and given the same height difference, the pressure differences in water are approximately equal at any height. </p> <div class="mw-heading mw-heading2"><h2 id="Hydrostatic_paradox">Hydrostatic paradox</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertical_pressure_variation&action=edit&section=2" title="Edit section: Hydrostatic paradox"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Diagram_illustrating_the_hydrostatic_paradox.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Diagram_illustrating_the_hydrostatic_paradox.svg/170px-Diagram_illustrating_the_hydrostatic_paradox.svg.png" decoding="async" width="170" height="195" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Diagram_illustrating_the_hydrostatic_paradox.svg/255px-Diagram_illustrating_the_hydrostatic_paradox.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/Diagram_illustrating_the_hydrostatic_paradox.svg/340px-Diagram_illustrating_the_hydrostatic_paradox.svg.png 2x" data-file-width="450" data-file-height="516" /></a><figcaption>Diagram illustrating the hydrostatic paradox</figcaption></figure> <p>The barometric formula depends only on the height of the fluid chamber, and not on its width or length. Given a large enough height, any pressure may be attained. This feature of hydrostatics has been called the <b>hydrostatic paradox</b>. As expressed by <a href="/wiki/W._H._Besant" title="W. H. Besant">W. H. Besant</a>,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><i>Any quantity of liquid, however small, may be made to support any weight, however large.</i></dd></dl> <p>The Flemish scientist <a href="/wiki/Simon_Stevin" title="Simon Stevin">Simon Stevin</a> was the first to explain the paradox mathematically.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> In 1916 <a href="/wiki/Richard_Glazebrook" title="Richard Glazebrook">Richard Glazebrook</a> mentioned the hydrostatic paradox as he described an arrangement he attributed to <a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Pascal</a>: a heavy weight <span class="texhtml mvar" style="font-style:italic;">W</span> rests on a board with area <span class="texhtml mvar" style="font-style:italic;">A</span> resting on a fluid bladder connected to a vertical tube with cross-sectional area α. Pouring water of weight <span class="texhtml mvar" style="font-style:italic;">w</span> down the tube will eventually raise the heavy weight. Balance of forces leads to the equation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W={\frac {wA}{\alpha }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>w</mi> <mi>A</mi> </mrow> <mi>α</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W={\frac {wA}{\alpha }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac152d55002bfdd1deaefdad921358f8d3ef8448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.424ex; height:5.343ex;" alt="{\displaystyle W={\frac {wA}{\alpha }}.}"></span></dd></dl> <p>Glazebrook says, "By making the area of the board considerable and that of the tube small, a large weight <span class="texhtml mvar" style="font-style:italic;">W</span> can be supported by a small weight <span class="texhtml mvar" style="font-style:italic;">w</span> of water. This fact is sometimes described as the hydrostatic paradox."<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Hydraulic_machinery#Force_and_torque_multiplication" title="Hydraulic machinery">Hydraulic machinery</a> employs this phenomenon to multiply force or torque. Demonstrations of the hydrostatic paradox are used in teaching the phenomenon.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="In_the_context_of_Earth's_atmosphere"><span id="In_the_context_of_Earth.27s_atmosphere"></span>In the context of Earth's atmosphere</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertical_pressure_variation&action=edit&section=3" title="Edit section: In the context of Earth's atmosphere"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Barometric_formula" title="Barometric formula">Barometric formula</a></div> <p>If one is to analyze the vertical pressure variation of the <a href="/wiki/Atmosphere_of_Earth" title="Atmosphere of Earth">atmosphere of Earth</a>, the length scale is very significant (<a href="/wiki/Troposphere" title="Troposphere">troposphere</a> alone being several <a href="/wiki/Kilometres" class="mw-redirect" title="Kilometres">kilometres</a> tall; <a href="/wiki/Thermosphere" title="Thermosphere">thermosphere</a> being several hundred kilometres) and the involved fluid (air) is compressible. Gravity can still be reasonably approximated as constant, because length scales on the order of kilometres are still small in comparison to Earth's radius, which is on average about 6371 km,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> and gravity is a function of distance from Earth's core.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Density, on the other hand, varies more significantly with height. It follows from the <a href="/wiki/Ideal_gas_law" title="Ideal gas law">ideal gas law</a> that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ={\frac {mP}{kT}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mi>P</mi> </mrow> <mrow> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {mP}{kT}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e89cfa1949f448fc5ce058bec945a20cfeb26d9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.569ex; height:5.343ex;" alt="{\displaystyle \rho ={\frac {mP}{kT}},}"></span> where </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">m</span> is average <a href="/wiki/Mass" title="Mass">mass</a> per air <a href="/wiki/Molecule" title="Molecule">molecule</a>,</li> <li><span class="texhtml mvar" style="font-style:italic;">P</span> is pressure at a given point,</li> <li><span class="texhtml mvar" style="font-style:italic;">k</span> is the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>,</li> <li><span class="texhtml mvar" style="font-style:italic;">T</span> is the <a href="/wiki/Temperature" title="Temperature">temperature</a> in <a href="/wiki/Kelvin" title="Kelvin">kelvins</a>.</li></ul> <p>Put more simply, air density depends on air pressure. Given that air pressure also depends on air density, it would be easy to get the impression that this was <a href="/wiki/Circular_definition" title="Circular definition">circular definition</a>, but it is simply interdependency of different variables. This then yields a more accurate formula, of the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{h}=P_{0}e^{-{\frac {mgh}{kT}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mi>g</mi> <mi>h</mi> </mrow> <mrow> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{h}=P_{0}e^{-{\frac {mgh}{kT}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fd2b81b7df6243ee362ad976464fb00bfe631ed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.974ex; height:4.176ex;" alt="{\displaystyle P_{h}=P_{0}e^{-{\frac {mgh}{kT}}},}"></span> where </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">P<sub>h</sub></span> is the pressure at height <span class="texhtml mvar" style="font-style:italic;">h</span>,</li> <li><span class="texhtml"><i>P</i><sub>0</sub></span> is the pressure at reference point 0 (typically referring to sea level),</li> <li><span class="texhtml mvar" style="font-style:italic;">m</span> is the mass per air molecule,</li> <li><span class="texhtml mvar" style="font-style:italic;">g</span> is the <a href="/wiki/Standard_gravity" title="Standard gravity">acceleration due to gravity</a>,</li> <li><span class="texhtml mvar" style="font-style:italic;">h</span> is height from reference point 0,</li> <li><span class="texhtml mvar" style="font-style:italic;">k</span> is the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>,</li> <li><span class="texhtml mvar" style="font-style:italic;">T</span> is the temperature in kelvins.</li></ul> <p>Therefore, instead of pressure being a <a href="/wiki/Linear" class="mw-redirect" title="Linear">linear</a> function of height as one might expect from the more simple formula given in the "basic formula" section, it is more accurately represented as an <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a> of height. </p><p>Note that in this simplification, the temperature is treated as constant, even though temperature also varies with height. However, the temperature variation within the lower layers of the atmosphere (<a href="/wiki/Troposphere" title="Troposphere">troposphere</a>, <a href="/wiki/Stratosphere" title="Stratosphere">stratosphere</a>) is only in the dozens of degrees, as opposed to their <a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">thermodynamic temperature</a>, which is in the hundreds, so the temperature variation is reasonably small and is thus ignored. For smaller height differences, including those from top to bottom of even the tallest of buildings, (like the <a href="/wiki/CN_Tower" title="CN Tower">CN Tower</a>) or for mountains of comparable size, the temperature variation will easily be within the single-digits. (See also <a href="/wiki/Lapse_rate" title="Lapse rate">lapse rate</a>.) </p><p>An alternative derivation, shown by the Portland State Aerospace Society,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> is used to give height as a function of pressure instead. This may seem counter-intuitive, as pressure results from height rather than vice versa, but such a formula can be useful in finding height based on pressure difference when one knows the latter and not the former. Different formulas are presented for different kinds of approximations; for comparison with the previous formula, the first referenced from the article will be the one applying the same constant-temperature approximation; in which case: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=-{\frac {RT}{g}}\ln {\frac {P}{P_{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>R</mi> <mi>T</mi> </mrow> <mi>g</mi> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>P</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=-{\frac {RT}{g}}\ln {\frac {P}{P_{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c37da458d1df5dd3764e29a611000fbab9532161" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.327ex; height:5.676ex;" alt="{\displaystyle z=-{\frac {RT}{g}}\ln {\frac {P}{P_{0}}}}"></span> where (with values used in the article) </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">z</span> is the elevation in meters,</li> <li><span class="texhtml mvar" style="font-style:italic;">R</span> is the <a href="/wiki/Gas_constant#Specific_gas_constant" title="Gas constant">specific gas constant</a> = <span class="nowrap"><span data-sort-value="7002287053000000000♠"></span>287.053 J/(kg K)</span></li> <li><span class="texhtml mvar" style="font-style:italic;">T</span> is the absolute temperature in kelvins = <span class="nowrap"><span data-sort-value="7002288150000000000♠"></span>288.15 K</span> at sea level,</li> <li><span class="texhtml mvar" style="font-style:italic;">g</span> is the acceleration due to gravity = <span class="nowrap"><span data-sort-value="7000980665000000000♠"></span>9.806<span style="margin-left:.25em;">65</span> m/s<sup>2</sup></span> at sea level,</li> <li><span class="texhtml mvar" style="font-style:italic;">P</span> is the pressure at a given point at elevation <span class="texhtml mvar" style="font-style:italic;">z</span> in <a href="/wiki/Pascal_(unit)" title="Pascal (unit)">Pascals</a>, and</li> <li><span class="texhtml"><i>P</i><sub>0</sub></span> is pressure at the reference point = <span class="nowrap"><span data-sort-value="7005101325000000000♠"></span>101,325 Pa</span> at sea level.</li></ul> <p>A more general formula derived in the same article accounts for a linear change in temperature as a function of height (lapse rate), and reduces to above when the temperature is constant: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z={\frac {T_{0}}{L}}\left(\left({\frac {P}{P_{0}}}\right)^{-{\frac {LR}{g}}}-1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>L</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>P</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>L</mi> <mi>R</mi> </mrow> <mi>g</mi> </mfrac> </mrow> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z={\frac {T_{0}}{L}}\left(\left({\frac {P}{P_{0}}}\right)^{-{\frac {LR}{g}}}-1\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9ddc756498a5afcfa6f9645d00b1f42d387a97" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:26.963ex; height:8.509ex;" alt="{\displaystyle z={\frac {T_{0}}{L}}\left(\left({\frac {P}{P_{0}}}\right)^{-{\frac {LR}{g}}}-1\right)}"></span> where </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">L</span> is the atmospheric lapse rate (change in temperature divided by distance) = <span class="nowrap"><span data-sort-value="3002349999999999999♠"></span>−6.5<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−3</sup> K/m</span>, and</li> <li><span class="texhtml"><i>T</i><sub>0</sub></span> is the temperature at the same reference point for which <span class="texhtml"><i>P</i> = <i>P</i><sub>0</sub></span></li></ul> <p>and the other quantities are the same as those above. This is the recommended formula to use. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertical_pressure_variation&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Barometer" title="Barometer">Barometer</a></li> <li><a href="/wiki/Hypsometric_equation" title="Hypsometric equation">Hypsometric equation</a></li> <li><a href="/wiki/Pascal%27s_barrel" class="mw-redirect" title="Pascal's barrel">Pascal's barrel</a></li> <li><i><a href="/wiki/Ruina_montium" title="Ruina montium">Ruina montium</a></i></li> <li><a href="/wiki/Pressure_gradient" title="Pressure gradient">Pressure gradient</a></li> <li><a href="/wiki/Siphon" title="Siphon">Siphon</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertical_pressure_variation&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/barfor.html">"The Barometric Formula"</a>.</cite><span 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(1966). <i>Fluid Mechanics</i>, 4th edition p.28, McGraw-Hill</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBesant1900" class="citation book cs1"><a href="/wiki/W._H._Besant" title="W. H. Besant">Besant, W. 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Springer Science & Business Media. p. 160. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-9400743458" title="Special:BookSources/978-9400743458"><bdi>978-9400743458</bdi></a>. <q>Stevin provides an original mathematical demonstration of the so-called hydrostatic paradox</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mechanization+of+Natural+Philosophy&rft.pages=160&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012-09-25&rft.isbn=978-9400743458&rft.aulast=Roux&rft.aufirst=Sophie&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertical+pressure+variation" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGlazebrook1916" class="citation book cs1"><a href="/wiki/Richard_Glazebrook" title="Richard Glazebrook">Glazebrook, Richard</a> (1916). <a rel="nofollow" class="external text" href="https://archive.org/stream/mechanicshydrost033294mbp#page/n55/mode/2up"><i>Hydrostatics: An elementary textbook, theoretical and practical</i></a>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p. 42 – via <a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Hydrostatics%3A+An+elementary+textbook%2C+theoretical+and+practical&rft.pages=42&rft.pub=Cambridge+University+Press&rft.date=1916&rft.aulast=Glazebrook&rft.aufirst=Richard&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fmechanicshydrost033294mbp%23page%2Fn55%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertical+pressure+variation" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreenslade,_Jr." class="citation web cs1">Greenslade, Jr., Thomas B. <a rel="nofollow" class="external text" href="https://physics.kenyon.edu/EarlyApparatus/Fluids/Hydrostatic_Paradox/Hydrostatic_Paradox.html">"Hydrostatic paradox"</a>. <a href="/wiki/Kenyon_College" title="Kenyon College">Kenyon College</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Hydrostatic+paradox&rft.pub=Kenyon+College&rft.aulast=Greenslade%2C+Jr.&rft.aufirst=Thomas+B.&rft_id=https%3A%2F%2Fphysics.kenyon.edu%2FEarlyApparatus%2FFluids%2FHydrostatic_Paradox%2FHydrostatic_Paradox.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertical+pressure+variation" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=6zeHWVUiXoc"><span class="plainlinks">Explanation</span></a> on <a href="/wiki/YouTube_video_(identifier)" class="mw-redirect" title="YouTube video (identifier)">YouTube</a></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.universetoday.com/26629/radius-of-the-earth/">"Radius of the Earth"</a>. 2 March 2009.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Radius+of+the+Earth&rft.date=2009-03-02&rft_id=http%3A%2F%2Fwww.universetoday.com%2F26629%2Fradius-of-the-earth%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertical+pressure+variation" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.splung.com/content/sid/2/page/gravitation/">"Newton's Law of Gravity"</a>. <i>www.splung.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">June 23,</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.splung.com&rft.atitle=Newton%27s+Law+of+Gravity&rft_id=http%3A%2F%2Fwww.splung.com%2Fcontent%2Fsid%2F2%2Fpage%2Fgravitation%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertical+pressure+variation" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110928003908/http://psas.pdx.edu/RocketScience/PressureAltitude_Derived.pdf">"A Quick Derivation relating altitude to air pressure"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://psas.pdx.edu/RocketScience/PressureAltitude_Derived.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2011-09-28<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-11-30</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+Quick+Derivation+relating+altitude+to+air+pressure&rft_id=http%3A%2F%2Fpsas.pdx.edu%2FRocketScience%2FPressureAltitude_Derived.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertical+pressure+variation" class="Z3988"></span></span> </li> </ol></div></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMerlino2003" class="citation web cs1">Merlino, Robert L. (2003). <a rel="nofollow" class="external text" href="http://www.physics.uiowa.edu/~rmerlino/6Fall06/6S06pp_L13.ppt">"Statics – Fluids at rest"</a><span class="reference-accessdate">. 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