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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.
Source publication
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...
Contexts in source publication
Context 1
... 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. ...
Context 2
... course, the magnitude of the forces will change continuously, but on average they should be zero in all directions at equilibrium. 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. ...
Context 3
... 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. ...
Citations
... All other gravitational effects will be neglected. The pressure difference between the bottom and the top of the gas is also considered negligible [9]. However, there is a difference between the horizontal and vertical position of the piston, but it is kinematic and not energetic. ...
The First Law of Thermodynamics represents the principle of energy conservation applied to the interaction between different macroscopic systems. The traditional mathematical description of the First Law (e.g.) is rather simplistic and lack universal validity, as it is only valid when several implicit assumptions are met. For example, it only considers mechanical work done associated with a change in volume of a system, but completely neglects other types of work. On the other hand, it employs the concept of entropy which is not only ambiguous but also implies only heat associated with a temperature difference, neglecting other types of heat transfer that may take place at mesoscopic and/or microscopic levels. In addition, it does not consider mass transfer effects. In the previous report of this series, a more general representation of the First Law is obtained considering different conditions and different types of interactions between the systems. In this report, the expression previously obtained is applied to different representative examples, involving macroscopic systems with no volume change, gas systems with volume change, and even a case where mass transfer between the systems takes place.
... Is Earth actually rotating? Why wind velocities don't match the expected speed of Earth, considering that momentum is not easily transmitted through gases [11], as it can be evidenced by their low viscosity values? Why doesn't Earth leave a tail of atmosphere like comets do? ...
This is a continuation of the fictional dialogue taking place by Descartes' hypothetical characters (Eudoxius, Epistemon and Polyander). In this opportunity, the discussion revolves around the common confusion between models and reality. Knowledge is built upon models, which are approximate representations of reality. However, it is practically impossible for us to determine the correctness of those models, and therefore, we will never know for sure which model is an accurate description of reality. Unfortunately, scientific models taught in schools and universities are tacitly assumed by most students (and educators) to be our reality, and this misconception limits scientific progress. For this reason, students, educators and scientists are invited to continuously and openly question all of our current scientific paradigms, even when those paradigms can be regarded as "universally accepted truths".
... Mathematical expressions for ideal gases are obtained [51]. In gases, forces (and particularly weight) cannot be directly transmitted, and thus, gases do not strictly follow Pascal's law [53]. This idea is also supported by experimental results [56]. ...
... This idea is also supported by experimental results [56]. An alternative analytical expression describing the effect of mass density and external acceleration on pressure changes in presented [53]. This result, based on the conservation of momentum, is used to derive a new barometric formula without recurring to the hydrostatic pressure assumption based on Pascal's law [58]. ...
... While this useful theorem is not new, it represented an important personal paradigm change as it was not previously known by the author. In 5 th place we find the analysis of the validity of Pascal's law in gases [53], reaching the (now evident to the author) conclusion that forces are not directly transmitted through gases as in the case of liquids and solids, and therefore Pascal's law is not valid for gases § § . 4 th place was given to the mechanistic model of liquid evaporation having a single parameter (cohesion temperature), representing an alternative to the empirical Antoine equation (with three parameters) for describing vapor pressure [93,99]. ...
The current report celebrates the 100 th report published by ForsChem Research, as well as the 7 th year since the beginning of the ForsChem Research Project. In this publication, a brief review of the evolution of ForsChem Research is presented, highlighting the most important contributions published in ForsChem Research Reports. A graphical bibliometric analysis is also included to illustrate the evolution of the works published during its first 7 years, and their impact as measured by ResearchGate (RG) stats. In addition, a selection of the author's top 10 favorite reports is presented. Finally, a brief outline is exposed about the plans for ForsChem Research in the future.
... Recently, an alternative derivation of the barometric formula was obtained without the hydrostatic assumption for the pressure of air [1], but taking into account that in reality Pascal's law is not valid for gases [2,3]. The barometric formula was obtained assuming air at steadystate, with a normal distribution of vertical molecular velocities at each altitude, and using only conservation equations (particularly, introducing the conservation of momentum instead of the hydrostatic pressure assumption). ...
The classical barometric formula used in atmospheric models is derived neglecting the presence of chemical reactions in the atmosphere. However, many chemical reactions are continuously taking place either promoted by sunlight or simply by the thermal motion of the molecules. In this report, the effect of chemical reactions on the barometric formula will be modeled and discussed. Such effect is not only related to individual molecular concentration profiles but also to thermal profiles when the heat of reaction is considered. The derivation of the model is based on a simple reversible chemical reaction, but it is also generalized for any arbitrary set of chemical reactions taking place in the system. Even under the steady-state assumption, the differential equations obtained do not provide a direct analytical solution and therefore, they must be numerically integrated. A particular example is presented for illustrating the model obtained but also the numerical solution method.
... Pascal's law was originally proposed for describing pressure differences in liquids [2], and has been shown unsuitable for gases [3,4]. In gases, the increase in pressure as the altitude decreases is not directly caused by the weight of the column of gas, as indicated by Pascal's law. ...
... Even more, the increase in atmospheric pressure close to the surface is even larger than the pressure increase expected by the increased weight of air. However, under certain conditions gases can behave approximately as a Pascal fluid [3]. ...
... Thus, it is not the purpose to change the mathematical model but only to change its interpretation. Furthermore, the larger rate of change in pressure compared to Pascal's law is consistent with previous theoretical developments [3,13] as well as experimental results [4]. ...
Barometric formulas are important mathematical equations used to understand and predict the behavior of the atmosphere pressure at different altitudes. Since the first development by Pierre-Simon de Laplace in the 18th century, the fundamental assumptions leading to barometric formulas have been considering air as an ideal gas at steady-state, and considering atmospheric pressure as a hydrostatic pressure following Pascal’s law. Being rigorous however, gases do not follow Pascal’s law since the molecules are on average so far from each other that they cannot transmit the weight of their neighboring molecules in the vertical direction. For this reason, a new barometric formula has been derived without recurring to the hydrostatic pressure assumption. Instead of Pascal’s law, the conservation of momentum is used to describe the effect of gravity on the vertical molecular density profile. Then, after determining the temperature profile (which can be derived by solving the energy conservation equation, or can be empirically obtained), the molecular density profile can be solved, and the vertical pressure profile can be directly obtained from the ideal gas equation. The barometric formula obtained, which is almost equivalent to the current barometric formula used by the standard atmospheric model (the US Standard Atmosphere of 1976), was tested considering a set of experimental barometric measurements reported from different locations worldwide. Even though only a slight difference is obtained, the new expression no longer requires assuming atmospheric pressure as hydrostatic. The wide success of previous barometric formulas can be explained by the fact that the pressure drop predicted by the conservation of momentum deviates by less than 4% from Pascal’s law. Finally, a multicomponent model of air was considered, which allows the estimation of atmospheric composition changes with altitude.
... A molecular model of an enclosed ideal gas subject to a constant external force [1] was used in a previous report [2] to show that Pascal's law does not describe the behavior of pressure differences in gases. Of course, several simplifying assumptions were necessary for obtaining the mathematical model. ...
... A graphical comparison of the model predictions assuming Pascal's law and the experimental results obtained is presented in Figure 7 where Pascal's law inadequacy can be clearly seen. Since the observed accelerations are smaller than those predicted using Pascal's law, it can be concluded that the actual air pressure difference is larger than that predicted by Pascal's law, in qualitative agreement with an alternative model described in a previous report [2]. ...
... The experimental results presented in this report indicate that Pascal's law does not correctly describe the pressure differences in atmospheric air, supporting the idea that pressure changes in the vertical direction are the result of differences in molecular velocities and densities, rather than the direct effect of the weight of the gas. The true behavior seems to be much more complicated than that predicted by Pascal's law [2]. ...
Pascal's law, while originally proposed for liquids, has usually been considered valid also for gases. However, the fact that the molecules in a gas are separated from each other beyond their spheres of action most of the time, does not allow the direct transmission of the weight of the gas in the vertical direction. Undoubtedly, the mass of gas and the gravitational acceleration influence the pressure of the gas, but not according to Pascal's law. Although a previously reported mathematical model of the motion of gas molecules supports this idea, an additional experimental verification is presented in this report. The validation of Pascal's law in gases is done considering the effect of differential air pressure on the acceleration of a body during free fall, particularly for low-density bodies. The free fall experiments were done considering four boxes (made of paper) of different sizes and densities, showing significantly lower free fall accelerations (p-value), compared to the predictions obtained assuming the validity of Pascal's law in gases.
... supercritical fluid, liquid or solid), suitable for the transmission of forces by contact through the entire body, whereas the other body must be elastic. In ideal gases, such transmission of forces is not possible [29], and therefore, energy transfer as work between two gas bodies cannot take place. Let us picture the interface between an ideal gas and a solid body, both at the same temperature ( Figure 22). ...
A dynamic model of a piston engine is presented based on the original engine proposed by Carnot in 1824. The dynamic model simulation can be used to analyze the Carnot cycle under carefully controlled ('reversible') conditions, as well as the operation of the engine under 'irreversible' conditions. The results obtained for the Carnot cycle are, in general, in agreement with the analytical results predicted by thermodynamic theory. The main difference observed is that the entropy change during a full cycle was negligible but not exactly zero. In fact, the entropy change at the end of the cycle can only be exactly zero if the whole process is adiabatic. In addition, different 'irreversible' scenarios were found that resulted in negligible total entropy change (considering also the surroundings), thus showing that it is not an exclusive property of 'reversible' systems. Different thermodynamic notions including energy, heat, work, reversibility, entropy, the Second Law of Thermodynamics, and engine efficiency, are discussed and explained, in order to get a better understanding from the results obtained.
