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Vol. 5, 2020-09

13/06/2020 ForsChem Research Reports Vol. 5, 2020-09 (1 / 13)

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Pascal’s Law in Gases

Hugo Hernandez

ForsChem Research, 050030 Medellin, Colombia

hugo.hernandez@forschem.org

doi:

Abstract

Pascal’s law or Pascal’s principle, enunciated almost 400 years ago, has been of utmost

importance in a wide variety of scientific and engineering disciplines. Currently physics

textbooks describe Pascal’s law as follows: “A change in the pressure applied to an enclosed

incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of

its container.” Such phenomenon can be explained based on the propagation of external forces

at a molecular level via intermolecular repulsion (following Newton’s third law of motion). In

the case of gases, such propagation of forces does not take place directly, although external

forces do influence the pressure of a gas by changing the momentum of the molecules hitting

the walls. There is, therefore, a relationship not necessarily identical to that stated in Pascal’s

law. However, under certain conditions, Pascal’s law remains as a fairly good approximation for

gases. For example, the International Standard Atmosphere model has assumed that the

atmosphere is an ideal gas following Pascal’s law, with satisfactory results.

Keywords

Blaise Pascal, Equivalent Pressure, Hydrostatic Equilibrium, Ideal Gases, Incompressible Fluids,

Intermolecular Forces, Non-ideal Gases, Repulsion

1. Introduction

In the 17th century, Galileo Galilei was asked by a fountain master of the time why a suction

pump (operating with the siphon principle) could not be used with water beyond a maximum

height of 32 feet. Galileo explained it using the notion of “resistenza del vacuo” (vacuum

resistance), derived from Aristotle’s horror vacui theory. Not completely convinced from his

own explanation, Galileo assigned the problem to one of his disciples: Evangelista Torricelli.

Pascal’s Law in Gases

Hugo Hernandez

ForsChem Research

hugo.hernandez@forschem.org

13/06/2020 ForsChem Research Reports Vol. 5, 2020-09 (2 / 13)

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Torricelli scaled-down this effect using mercury instead of water (almost 13.6 times heavier),

finding a maximum siphon height of just 760 mm. His experiments, which rejected Galileo’s

original hypothesis by demonstrating that air (and not vacuum) exerted pressure on the liquid,

also gave rise to the invention of the barometer around 1643. In 1647, Blaise Pascal, a French

mathematician, religious philosopher and physicist, became aware of Torricelli’s experiments

and decided to reproduce and improve them [1].

Pascal results and findings can be found in different publications, including his “Traités de

l’Equilibre des Liqueurs, et de la Pesanteur de la Masse de l’Air” (“Treatises on the equilibrium of

liquids, and on the gravity of the mass of air”). The first chapter of his treatise on the

equilibrium of liquids was entitled “Que les Liqueurs pesent suivant leur hauteur” (“That liquids

weight according to their height”) [2]. This statement, which will later be known as Pascal’s

law, is the principle of operation of the hydraulic press, invented more than a century later [3],

and other machines.

Mathematically, Pascal’s law can be expressed as follows

§

:

(1.1)

where is the difference in pressure between two different heights of a fluid (also

known as hydrostatic pressure), is the mass density of the fluid, is the gravitational

acceleration, and is the corresponding height difference. Please note the change

in the order of the pressure and height differences. Considering height as positive in the

direction from the ground to the atmosphere, and considering differential changes in height,

the mathematical (differential) representation of Pascal’s law becomes:

(1.2)

A modern interpretation of Pascal’s law is [4]: “A change in the pressure applied to an enclosed

incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of

its container.” Mathematically this is:

(1.3)

§

Pascal originally considered the weight (related to the pressure at the bottom) of the same liquid when

the level in the container changes. Later it was extended to consider the pressure of a column of liquid at

different positions, resulting in the famous expression (Eq. 1.1).

Pascal’s Law in Gases

Hugo Hernandez

ForsChem Research

hugo.hernandez@forschem.org

13/06/2020 ForsChem Research Reports Vol. 5, 2020-09 (3 / 13)

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The pressure at a certain depth below the surface of a liquid is the sum of the external

pressure and the hydrostatic pressure

**

. Thus, a change in the external

pressure acting on the contained incompressible liquid, results in a pressure change at depth :

(1.4)

Pascal’s law can be used in a wide variety of situations. It is the operating principle of different

hydraulic machines and hydraulic components, and is also useful in the design of liquid

containers, dams, ships, submarines, and scuba diving equipment, just to mention a few.

Pascal’s law, along with the ideal gas equation, is a key principle of the International Standard

Atmosphere (ISA) model [5]. In this model, the change in temperature with decreasing altitude

(lapse rate) is assumed constant for each layer of the atmosphere, and the corresponding

pressure profile is obtained assuming that the atmosphere is at hydrostatic equilibrium. Thus,

from Pascal’s law (Eq. 1.2) and the ideal gas equation:

(1.5)

where is the average molecular mass of air, is Boltzmann constant, and is the absolute

temperature. Since is assumed to change linearly with respect to altitude (constant lapse

rate), Eq. (1.5) becomes:

(1.6)

where is the absolute temperature on the ground, and is the constant lapse rate (at the

troposphere). Solving Eq. (1.6) yields:

(1.7)

The air in this model is also assumed to be dry, dust-free, and at rest with respect to Earth. This

hypothetic approximate model of the atmosphere has been used for calibrating pressure

altimeters, and for the design of rockets and aircrafts [6].

**

The depth of liquid below the surface is related to the height by: , where is the level of

liquid in the container.

Pascal’s Law in Gases

Hugo Hernandez

ForsChem Research

hugo.hernandez@forschem.org

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However, considering that Pascal investigated liquids, and also from the modern definition of

Pascal’s law we realize that it is limited to enclosed incompressible fluids (which exclude gases),

then the following question emerges: Is it safe to extrapolate Pascal’s law to gases, and

particularly, to ideal gases? Please notice that the ISA model also assumes that air is an ideal

gas, and also that the atmosphere is not enclosed or contained by rigid walls.

2. Molecular Interpretation of Pascal’s Law in Liquids

Before discussing the applicability of Pascal’s law in gases, this principle will be analyzed from a

molecular perspective. Let us consider an incompressible fluid (liquid) confined in a

impermeable solid container with constant cross-section as depicted in Figure 1. Gravity, of

course, is acting in the vertical downward direction. A specific part of the container is zoomed

in, only to find that the fluid is composed of molecules (represented by spheres). Assuming

that the system is at rest (hydrostatic equilibrium), the average net force acting on the

molecules of the fluid is zero. Since gravity is permanently acting on each molecule, such

equilibrium of forces is only possible if the intermolecular forces exactly compensate gravity, as

illustrated in Figure 2 (considering only two layers of molecules). Of course, the magnitude of

the forces will change continuously, but on average they should be zero in all directions at

equilibrium.

Figure 1. Schematic representation of an incompressible fluid (blue) confined in a cubic

container (only edges shown in black). Left: Macroscopic scale. Right: Molecular scale.

Pascal’s Law in Gases

Hugo Hernandez

ForsChem Research

hugo.hernandez@forschem.org

13/06/2020 ForsChem Research Reports Vol. 5, 2020-09 (5 / 13)

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Figure 2. Forces acting on the molecules of a liquid (cyan circles), considering only two layers of

molecules. Green arrows: Gravity. Red arrows: Intermolecular repulsion forces. Black arrows:

Normal force (repulsion force between the fluid molecule and the solid wall). Blue arrows:

Additional external forces. Left plot: No additional external forces considered. Right plot:

Additional external forces considered.

The molecules at the top of Figure 2 compensate the gravitational force acting on them due to

the repulsion forces of the molecules below them. On the other hand, the molecules at the

bottom of the container compensate the gravitational force acting on them plus the repulsion

forces of the upper molecules, thanks to the repulsion of the solid wall. Now, since at

equilibrium the vertical components of the repulsive forces at the top molecules correspond to

the gravitational force () but with opposing direction, then by Newton’s third law (action-

reaction) the repulsive force perceived by the molecules at the bottom will correspond to the

weight of the upper molecules. Adding the repulsive forces to their own gravitational force,

the net force acting on the solid wall is the sum of all the molecules above it. By adding more

layers of molecules, the result is the same: The gravitational force (weight) of the upper

molecules is transmitted to the lower molecules as a result of intermolecular repulsions.

Mathematically, the magnitude of the total force acting on the solid wall at the bottom

() would be expressed as follows:

(2.1)

where is the number of molecules in the container. Now, since the mass density of the fluid

is given by:

(2.2)

Pascal’s Law in Gases

Hugo Hernandez

ForsChem Research

hugo.hernandez@forschem.org

13/06/2020 ForsChem Research Reports Vol. 5, 2020-09 (6 / 13)

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where is the volume of incompressible fluid in the container, is the cross-section area at the

bottom of the container, and is the height of fluid, then Eq. (2.1) becomes:

(2.3)

Finally, since the pressure is defined as force per unit area , we get:

(2.4)

This expression closely resembles Pascal’s law, but we are still missing something. The

molecules at the top of the fluid can also be exposed to additional (external) forces as can be

seen in the right scheme in Figure 2. In this case, the external force will increase the repulsion

forces between fluid molecules, and also will increase the normal repulsion forces at the

bottom wall. In other words, the external force propagates in the direction of the force by

increasing the magnitude of the intermolecular repulsive forces found along the path of the

force. This effect is described by the modern formulation of Pascal’s law, when using the

expression ‘undiminished transmission of the changes in pressure’ (or changes in external

forces). Furthermore, the molecules close to the side walls will also be pushed towards those

walls, also resulting in a lateral (isotropic) transmission of the pressure.

This analysis could be easily extended to solid systems, concluding that Pascal’s law should also

be valid for solids. Of course, the pressure inside most solid systems cannot be easily

determined, but in certain solid systems it is possible (imagine for example particles of sand or

any other solid material). The main differences between solid and liquid systems at the

molecular level are that the motion of molecules at the solid state is more constrained than at

the liquid state, and that molecular contact of liquids with the walls is more homogeneous.

However, the transmission of force by intermolecular repulsion also takes place in solids. In the

following section, the case of gas systems will be considered.

3. Pascal’s Law in Gases

3.1. Molecular Analysis of Pascal’s Law in Gases

Let us now imagine a similar situation for a container filled with a gas. The molecular

representation at the bottom wall is presented in Figure 3. Given that gas molecules are less

crowded, repulsion forces are only present when molecules collide between them, or when

they hit the walls.

Pascal’s Law in Gases

Hugo Hernandez

ForsChem Research

hugo.hernandez@forschem.org

13/06/2020 ForsChem Research Reports Vol. 5, 2020-09 (7 / 13)

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Figure 3. Forces acting on the molecules of a gas (purple circles). Green arrows: Gravity. Red

arrows: Intermolecular repulsion forces. Black arrows: Normal force (repulsion force between

the fluid molecule and the solid wall). Blue arrow: Additional external force.

First, we can see that repulsive forces emerging during molecular collisions have no direct

effect on the pressure of the wall, since those forces cannot propagate at such low densities.

The same thing occurs with external forces, they are not propagated. Since equilibrium (local

equilibrium at the molecular level) cannot be reached, those forces will have an effect on the

speed and direction of the molecules. Now, a small fraction of molecules will be hitting the wall

at a certain instant. The number of molecules hitting the wall is the molecular flux, which mainly

depend on the molecular density, but also on the temperature of the system and the relative

macroscopic motion of the gas [7]. In our case, since the gas is completely confined there is no

relative macroscopic motion of the gas, or in other words, the gas is static or at rest. On the

other hand, the resulting normal repulsive force at any wall will be determined by the

momentum of the collision. Faster and heavier molecules will result in larger repulsion forces,

thus increasing the pressure at the walls. Thus, mass and gravity have an important influence

on the pressure at the walls, but not related to the transmission of the weight through the gas,

but as a result of the increased molecular momentum when colliding with the walls.

The previous analysis clearly indicates that Pascal’s law cannot be directly extended to gas

systems. This does not necessarily mean that density, gravity and height of gas have no effect

on pressure.

3.2. Ideal Gas under Constant Force Model

Let us consider an ideal gas system contained in an impermeable thermally-insulated container,

subject to a constant force. The cross-section of the container, perpendicular to the external

Pascal’s Law in Gases

Hugo Hernandez

ForsChem Research

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13/06/2020 ForsChem Research Reports Vol. 5, 2020-09 (8 / 13)

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force, is constant. Before the force is applied, the distribution of molecular positions is

assumed uniform, and the distribution of molecular speeds is assumed to follow the Maxwell-

Boltzmann distribution. For this particular system, it is possible to approximately estimate the

local density, temperature and pressure profiles in the direction of the external force that will

be obtained at equilibrium [8]. Particularly, the local (equivalent

††

) pressure is given by the ideal

gas equation:

(3.1)

where and are the local equilibrium molecular density and absolute temperature

at depth from the top of the container, is the total height of the container, is the initial

absolute temperature of the system, and is the molecular diameter. The equivalent

pressure at a certain height was found to be identical to the pressure exerted on the walls at

the same height (pressure isotropy).

Additionally, the local equilibrium temperature and local molecular density are given by [8]:

(3.2)

(3.3)

Where is the overall molecular density in the container.

Thus, the change in local pressure with respect to depth will be:

(3.4)

where

is known as the lapse rate.

††

The term equivalent pressure indicates that forces (pressure) only appear when a physical barrier is

imposed on the system (e.g. the walls of the container or the pressure measurement device). The

internal pressure of an ideal gas is zero (or negligible), since no significant intermolecular forces are

present [7].

Pascal’s Law in Gases

Hugo Hernandez

ForsChem Research

hugo.hernandez@forschem.org

13/06/2020 ForsChem Research Reports Vol. 5, 2020-09 (9 / 13)

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Given the complexity of Eq. (3.2), instead of presenting a general expression for the lapse rate

let us consider two different scenarios: 1) Lab-scale dimensions

, and 2) Atmosphere-

scale dimensions

. In the first scenario, a very rough approximation of the lapse rate

is:

(3.5)

whereas for the second case we have:

(3.6)

Similarly, the change in molecular density with respect to depth will be given by:

(3.7)

(3.8)

The local change in pressure then becomes:

(3.9)

Pascal’s Law in Gases

Hugo Hernandez

ForsChem Research

hugo.hernandez@forschem.org

13/06/2020 ForsChem Research Reports Vol. 5, 2020-09 (10 / 13)

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(3.10)

Clearly, the rate of change in the pressure of ideal gases under an external force with respect

to depth (or with respect to height by reversing the sign) is not constant, as in the case of

liquids (Pascal’s law). However, gravity, density and depth show an important effect on the

rate of pressure change.

A graphical comparison between the results obtained for air using Eq. (3.4) and those expected

using Pascal’s law (Eq. 1.5) is presented in Figure 4. Different values of

are considered. For

greater accuracy, especially at intermediate values of

, instead of using the analytical

approximations given in Eq. (3.9) and (3.10), the rate of pressure change is obtained by

numerically differentiating Eq. (3.2).

Figure 4. Relative rate of pressure change

as a function of the relative depth

for

different values of the ratio

. Pascal’s law (black solid line) is included for comparison.

Even considering that the results obtained by the ideal gas under constant force model are just

approximate, it is possible to conclude that Pascal’s law is, in general, not valid for ideal gases.

For relatively small containers, the pressure changes are small but much larger than those

predicted by Pascal’s law. However, in the middle section of the container the rate of pressure

Pascal’s Law in Gases

Hugo Hernandez

ForsChem Research

hugo.hernandez@forschem.org

13/06/2020 ForsChem Research Reports Vol. 5, 2020-09 (11 / 13)

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change remains relatively constant, resembling Pascal’s law but with a proportionality factor.

For very large containers (or large systems), the pressure changes are large but usually smaller

than those predicted by Pascal’s law. For intermediate values of

, Pascal’s law

can be considered a fairly good approximation, particularly in the middle and top sections of

the container.

In 1965, Yalamov and Derjagin [9] found, using the mathematical approach proposed by James

Clerk Maxwell [10], that pressure anisotropy emerges in gases formally violating Pascal’s law.

However, they did not observe anisotropy experimentally, attributing this effect to an

incomplete mathematical formulation of Maxwell [11]. Their experiments were performed in

very small systems ( in height), subject to large temperature gradients

(). They experimentally validated the pressure anisotropy of gases, one

component of Pascal’s law, but their results did not allow verifying the validity of Eq. (1.5).

3.3. Pascal’s Law in Non-ideal Gases

As the density of a gas increases, the average distance between molecules is reduced. At some

point, when the average intermolecular distance approaches the radius of action of the

molecules, non-ideal gas behavior emerges [12]. At this point, a weak but permanent

interaction between molecules will be present, which will partially transfer gravitational and

other external forces. Thus, as the average intermolecular distance is reduced in the non-ideal

gas, it is expected that the rate of pressure change approaches Pascal’s law. Non-ideal gas

behavior can be considered as an intermediate state between ideal gases and liquids. At a

specific instant, some molecules will behave as ideal gases and some other almost as liquids

(transferring forces via intermolecular repulsion), depending on the local molecular densities. If

we consider that a fraction of molecules behave as ideal gas molecules while the remaining

show a liquid-like behavior, then the rate of pressure change of a non-ideal would be

approximately given by:

(3.11)

where

is given by Eq. (3.4).

Pascal’s Law in Gases

Hugo Hernandez

ForsChem Research

hugo.hernandez@forschem.org

13/06/2020 ForsChem Research Reports Vol. 5, 2020-09 (12 / 13)

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4. Conclusion

Pascal’s law is a physical principle used for many important applications in science and

engineering, indicating that external forces (or pressure) applied to contained incompressible

fluids are transmitted or propagated through the fluid. This occurs because external forces are

balanced by intermolecular repulsive forces.

In gases, however, forces (and particularly weight) cannot be directly transmitted as predicted

by Pascal’s law. Of course, the mass density of the gas and the external acceleration determine

the pressure changes in the system because they influence the momentum of the molecules

colliding with the walls, but not as the result of intermolecular force propagation. Thus, gases

do not strictly follow Pascal’s law.

The International Standard Atmosphere model, while being a good approximation for

describing the behavior of the atmosphere based on physical principles such as the ideal gas

equation and Pascal’s law, cannot be considered a rigorous first-principles model because using

Pascal’s law in this system might be phenomenologically erroneous.

Acknowledgments

The author gratefully acknowledges Prof. Jaime Aguirre (Universidad Nacional de Colombia)

for fruitful discussions and for reviewing and proof-reading the manuscript.

This research did not receive any specific grant from funding agencies in the public,

commercial, or not-for-profit sectors.

References

[1] Loeffel, H. (1987). Blaise Pascal 1623–1662. Birkhauser Verlag, Basel.

[2] Pascal, B. (1663). Traites de l'equilibre des liqueurs, et de la pesanteur de la masse de l'air.

Chez Guillaume Desprez, Paris.

[3] Bramah, J. (1795). Obtaining and Applying Motive Power. British Patent No. 2045.

[4] Walker, J. (2018). Fundamentals of Physics Halliday & Resnick. Extended 11th Ed. John Wiley

& Sons, Hoboken. p. 393.

[5] ISO 2533:1975 (1975). Standard Atmosphere. International Organization for Standardization.

Pascal’s Law in Gases

Hugo Hernandez

ForsChem Research

hugo.hernandez@forschem.org

13/06/2020 ForsChem Research Reports Vol. 5, 2020-09 (13 / 13)

www.forschem.org

[6] Cavcar, M. (2000). The international standard atmosphere (ISA). Anadolu University,

Turkey. http://www.academia.edu/download/52648552/ISAweb.pdf

[7] Hernandez, H. (2019). Calculation of Molecular Fluxes and Equivalent Pressure in Ideal

Gases. ForsChem Research Reports, 4, 2019-03. doi: 10.13140/RG.2.2.35898.44483.

[8] Hernandez, H. (2020). Effect of External Forces on the Macroscopic Properties of Ideal

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[9] Yalamov, Ya. I. & Deryagin, B. V. (1965). К ВОПРОСУ О ТЕМПЕРАТУРНЫХ НАПРЯЖЕНИЯХ В

НЕОДНОРОДНО НАГРЕТЫХ ГАЗАХ (Stresses due to temperature in unevenly heated gases).

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[10] Maxwell, J. C. (1879). VII. On stresses in rarified gases arising from inequalities of

temperature. Philosophical Transactions of the royal society of London, (170), 231-256.

[11] Deryagin, B. V. & Rabinovich, Ya. I. (1965). ЭКСПЕРИМЕНТАЛЬНАЯ ПРОВЕРКА

ПРИМЕНИМОСТИ ЗАКОНА ПАСКАЛЯ В НЕОДНОРОДНО НАГРЕТЫХ ГАЗАХ (Experimental

test of the validity of Pascal’s law in unevenly heated gases). Dokl. Akad. Nauk SSSR, 162 (1),

50–53.

[12] Hernandez, H. (2017). Clausius’ molecular sphere of action in crowded systems: Non-ideal

gas behavior. ForsChem Research Reports, 2, 2017-11. doi: 10.13140/RG.2.2.36384.07681.