Equation barometric

For a tall column containing an ideal gas with a molar mass M, in kg.mol , it is important to take into account the volume compressibility of the gas, i.e., the variation of its mass density with absolute pressure. In the column, the pressure variation, dP, in Pa is given by  

The negative sign indicates that the pressure increases from the top of the column to the bottom as molecules lower down are affected by the mass of those higher up. Replacing the density of the ideal gas using the equation obtained at the beginning of Section 19.1.9, we obtain the following  

After integration, the pressure in the gas column is expressed as a direct function of the elevation, absolute temperature, and molar mass  

This equation is called the barometric equation by geophysicists (i.e., aerologists and meteorologists) because it provides the absolute pressure in an air column for a given geometric elevation (i.e., altitude, Z), and in this particular case the reference elevation is taken equal to 0 meters (i.e., on Earth, the sea level is the datum plane). Usually, the factor denoted H = RT/Mg is named the scale height and it is expressed in m. It corresponds to the elevation at which the absolute gas pressure is divided by the Naperian base e = 2.718281828... 

Example If we assume, in a first approximation, air to be an ideal gas, having the average molar mass M = 28.930 x 10 kg.moT , a mass density / = 1.293 kg.m at T= 273.15 K, the scale height is equal to about 7.986 km. This means that for an altitude equal to that of Mount Everest (8.846 km), the absolute pressure is roughly a third of pressure exerted at sea level  

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Note that this expression is equivalent to the barometric formula which gives the variation of atmospheric pressure ( c) with elevation (oc r). A first-order dependence on the distance variable holds in the barometric equation, since the acceleration is constant in this case. 

Were it not for the never-ending, gentle tussle between gravity and diffusion, our planet would not have an atmosphere, nor would we be here to reflect on it The barometric equation, which describes this balance of power between the above two well-known phenomena, is derived in most introductory physical chemistry books and is mentioned in the closing paragraph of this chapter. There are many more life-sustaining processes that are... 

This familiar equation gives the variation of barometric pressure with elevation and is known as the barometric equation. Once again we are reminded of the connection between the material of this chapter and kinetic molecular theory. 

Describe the relation between sedimentation/diffusion equilibrium and the barometric equation. 

The pressure P due to the atmosphere is related to the height h of the mercury column in a Torricelli103 barometer, by the so-called barometric equation ... 

Using the barometric equation, compute the height at which 50 percent of 0.05-p.m unit-density spheres would be suspended by molecular impacts. 

If the test is being conducted at elevation other than zero or one atmosphere of barometric pressure, me [C J obtained from Table 9.1 must be corrected for me pressure of me test. Thus, during me performance of the experiment, the barometric pressure at the location should be recorded. Also, the barometric pressure at the test location may be obtained using the barometric equation. This equation relates the barometric pressure wim altitude z in me troposphere, and from fluid mechanics, this equation is... 

The Boltzmann distribution describes the relative population of molecules at different altitudes in a column of air above the surface of the earth due to the differences in their gravitational potential energy at these altitudes. The most convenient form of this distribution—called the barometric equation—relates the pressure Fy in the column at altitude h to its value Pq t the surface of the earth as Fy = Pq exp[—TfgP/RT], where M, is the molar mass of the gas and g is the acceleration of gravity. Calculate the pressure at an altitude of 1 km for a gas with average molar mass 29 g mol when the temperature is 298 K and the pressure at the surface of the earth is 1 atm. 

In a mixture of gases like the troposphere, Dalton s law of partial pressure (Box 3.1) is obeyed. This means that individual gases in the atmosphere will decline in pressure at the same rate as the total pressure. This can all be conveniently represented by the barometric equation ... 

BAROMETRIC EQUATION. For an ideal gas the density and pressure are related by the equation... 

This is the basic equation of fluid statics, also called the barometric equation. It is correct only if there are no shear stresses on the vertical faces of the cube in Fig. 2.1. If there are such shear stresses, then they may have a component in the vertical direction, which must be added to the sum of forces in Eq. 2.1. For simple newtonian fluids, shear stresses in the vertical direction can exist only if the fluid has a different vertical velocity on one side of the cube from that on the other side (see Eq. 1.5). Thus this equation is correct if the fluid is not moving at all, which is the case in fluid statics, or if it is moving but only in the X and y directions, or if it has a uniform velocity in the z direction. In this chapter, we apply it only when a fluid has no motion relative to its container or to some set of fixed coordinates. In later chapters, we apply it to flows in which there is no motion in the z direction or a motion with a uniform z component. 

The barometric equation tells the change in pressure with distance upward, where upward is opposite to the direction of gravity and is called z. If we want to know the change in pressure with distance in some other, nonvertical direction, call it direction a, then we can write... 

This expression is nsed in the example of a gas colnmn submitted to gravity (see case study K3 Barometric Equation in Chapter 13). 

K3 Barometric Equation Gravitation Hydrodynamics Physical chemistry Thermics External 698... 

In this formula, the distance r between the mass centers has been replaced by the sum of the radius of the reference body (earth) and of the altitude z. The altitude origin (z = 0) corresponds to the surface of this body. Equation K3.14 is therefore a general model of a column of material in a Newtonian (linear medium) gravity field. In order to retrieve the barometric Equation K3.1, a certain number of approximations must be made. The first one is to assume a constant gravity field, thus allowing the writing of the gravitational capacitive relationship in Equation K3.14 as... 

The insertion of this expression into Equation K3.7 leads directly to the barometric Equation K3.1... 

K3 Barometric Equation M = = -4- n Vq UjMs = = -pn Gravitation/physical chemistry Invariable... 

Among the various case studies of energy coupling presented, it is worth discussing the case of multiple couplings in the same systan, as in the case of the ideal gas (case study K2) and of the barometric equation (case study K3). In the case of the ideal gas, there is equality between the two coupling energies ... 

In the case of the barometric equation, which is based on an ideal gas, the same equality as in Equation 13.22 is met, but without including the gravitational energy which is independently coupled to the physical chemical energy ... 

K3 Barometric Equation Coupling with hydrodynamics, thermics. C... 

Barometric Equation Kinetic of Gases Euler Equation... 

Newtonian Gravitation Gravitational Force Barometric Equation... 

As a general rule, the barometric equation is useful when dealing with a huge column of gas as in oil drilling or geothermy. However, for usual chemical engineering calculations, it is a common agreement to consider a constant pressure in the overall gas column. 

An important equation relating to fluid statics is the barometric equation... 

We can also apply the barometric equation to directions other than those that are directly vertical. In such cases we use trigonometry to correct the equation... 

In using the barometric equation, we also must consider the usage of the terms absolute, gauge, and atmospheric pressure. The interrelation between these is given in equation (2-3) ... 

In this example we will use a detailed approach in order to demonstrate how the barometric equation works. Further more, the units used will be English in order to help us illustrate some important facts relating to units. 

The overall pressure differential in the system shown in Figure 2-4 is (Fi — P5). In order to derive the expression for it, let us use the barometric equation step by step. 

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