# Osmosis and hydrostatic pressure

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I'm confused about the role of hydrostatic pressure compared to osmotic pressure.

Q1:If I have a U-tube with a membrane permeable only to water molecules and equal volumes of water on either side but only 1 side (side B) has NaCl, the osmotic pressure would cause water to move from side A to side B,correct ?

Q2.But then hydrostatic pressure would cause water to move back to side A. So, the water would move from side A to side B until the effect caused by the hydrostatic pressure = effect caused by osmotic pressure ?

Q3. The last statement wouldn't be correct If I said that "water moves until the hydrostatic pressure=osmotic pressure" would it ?

and lastly, My friend said the water would move until the hydrostatic pressure on both sides was equal Q4. If water is moving from side A to side B then the we have more water molecules at side B, how would side A ever reach the hydrostatic pressure at side B ? Do I have a misunderstanding in the concept of hydrostatic pressure ? In this context, I understand that it is the pressure exerted by the water molecules on the selectively permeable membrane.

The more I google hydrostatic pressure the more lost I get because all sources seem to explain in terms of equations and physics and I'm only taking this for an introductory course in physiology.

Osmosis is defined as the flow of water/solvent molecules through a semipermeable membrane from a region of low to high solute concentration, until equilibrium is established.

To counter osmotic flow, some pressure must be applied to the solution in order to prevent pure solvent from going through the semipermeable membrane separating the two liquids; this is known as the osmotic pressure.

The osmotic pressure is the pressure required to counter, not sustain, osmosis.

The osmotic pressure can be approximated by using the following formula: $$Pi = i M R T$$ .

U- Tube showing osmotic pressure. On the left side of the U-tube is an aqueous solution, and on the right side is pure water. The pure water is trying to dilute the solution by travelling through the semipermeable membrane. Eventually the added weight of the extra water on the left causes enough pressure to stop osmosis.

Osmotic pressure is the pressure that needs to be applied to a solution to prevent the inward flow of water across a semipermeable membrane. Osmotic pressure can also be explained as the pressure necessary to nullify osmosis. One way to stop osmosis is to increase the hydrostatic pressure on the solution side of the membrane; this ultimately squeezes the solvent molecules closer together, increasing their “escaping tendency.” The escaping tendency of the solution can be raised until it eventually equals that of the molecules in the pure solvent; at this point, osmosis will cease. The osmotic pressure is the pressure required to achieve osmotic equilibrium.

Osmotic pressure. Osmotic pressure is the pressure required to stop osmosis.

The osmotic pressure (II) of an ideal solution can be approximated by the Morse equation:

$$Pi = i M R T$$

Here, i is the van 't Hoff factor, M is the molarity of the solution, R is the gas constant, and T is the absolute temperature in Kelvin. We can see from this equation that the amount of solute present in the solution will directly affect the osmotic pressure of the system.

Example

What is the osmotic pressure of a 1.35 M solution of NaCl at 25 $$^circ$$C?

First, fill in all of the necessary information, and then solve:

i = 2 (NaCl breaks into two particles)

M = 1.35 $$frac{moles}{L}$$

R = 0.0821 $$frac{L imes atm}{K imes mol}$$

T = 25 $$^circ$$C + 273 = 298 K

$$Pi = 2 imes 1.35 imes 0.0821 imes 298$$

$$Pi = 66.1 atm$$

## Key Takeaways

• Osmosis is the net movement of solvent molecules through a partially permeable membrane into a region of higher solute concentration in order to equalize the solute concentrations on the two sides.
• Osmosis provides the primary means by which water is transported into and out of cells.
• Osmoregulation is the homeostasis mechanism of an organism to reach balance in osmotic pressure.
• If the medium is hypotonic, the cells will gain water through osmosis.
• If the medium is hypertonic, the cells will lose water through osmosis.

## Difference Between Hydrostatic Pressure and Osmotic Pressure

Pressure is defined as the force per unit area applied in a direction perpendicular to the object. Hydrostatic pressure is the pressure experienced by a point inside the fluid. Osmotic pressure is the pressure that is needed to stop the fluid transfer of a semi permeable membrane. These concepts play a vital role in fields such as hydrostatics, biology, plant sciences and many other fields. It is vital to have a clear understanding in these concepts in order to excel in such fields. In this article, we are going to discuss what osmotic pressure and hydrostatic pressure are, the definitions of these two, similarities between hydrostatic pressure and osmotic pressure and finally the difference between osmotic pressure and hydrostatic pressure.

What is Hydrostatic Pressure?

The pressure of a static fluid is equal to the weight of the fluid column above the point the pressure is measured. Therefore, the pressure of a static (non-flowing) fluid is dependent only on the density of the fluid, the gravitational acceleration, the atmospheric pressure and the height of the liquid above the point the pressure is measured. The pressure can also be defined as the force exerted by the collisions of particles. In this sense, the pressure can be calculated using the molecular kinetic theory of gasses and gas equation. The term “hydro” means water and the term “static” means non-changing. This means hydrostatic pressure is the pressure of the non-flowing water. However, this is also applicable to any fluid including gasses. Since the hydrostatic pressure is the weight of the fluid column above the measured point it can be formulated using P= hdg, where P is the hydrostatic pressure, h is the height of the surface of the fluid form the measured point, d is the density of the fluid, and g is the gravitational acceleration. The total pressure on the measured point is the unison of the hydrostatic pressure and the external pressure (i.e. atmospheric pressure) on the fluid surface.

What is Osmotic Pressure?

When two solutions having different solute concentrations are divided by a semi permeable membrane, the solvent at the low concentrated side tends to move to the high concentration side. Imagine a balloon made of the semi permeable membrane filled with high concentration solution submerged inside the low concentrated solvent. The solvent will transfer to the inside of the membrane. This will cause the pressure of the inside of the membrane to rise. This risen pressure is known as the osmotic pressure of the system. This is a vital mechanism in transferring water to the inside of the cells. Without this mechanism, even trees cannot survive. The inverse of osmotic pressure is known as water potential, which is the tendency of the solvent to stay in the solution. Higher the osmotic pressure, lower will be the water potential.

What is the difference between Hydrostatic Pressure and Osmotic Pressure?

• Hydrostatic pressure is observed in any fluid, which is not moving. Osmotic pressure is only present in specific systems where the solution and the solvent are separated by a semi permeable membrane.

• Osmotic pressure cannot occur only with a pure fluid. Two different concentrated solutions are required for osmotic pressure. Hydrostatic pressure can occur only with one fluid.

## Osmosis and hydrostatic pressure - Biology

A solution is defined as a homogeneous mixture of both a solute and solvent. Solutions generally have different properties than the solvent and solute molecules that compose them. Some special properties of solutions are dependent solely on the amount of dissolved solute molecules, regardless of what that solute is these properties are known as colligative properties.

Osmosis is defined as the net flow or movement of solvent molecules through a semipermeable membrane through which solute molecules cannot pass. If a solution consisting of both solute and solvent molecules is placed on one side of a membrane and pure solvent is placed on the other side, there is a net flow of solvent into the solution side of the membrane.

Imagine osmosis taking place in an upright U-tube. The height of the solution will continue to increase due to a net flow of solvent until the added pressure of the height will cause the flow of solution to stop. The height difference between the two sides can be be converted into pressure to find the osmotic pressure exerted on the solution by the pure solvent.

U-Tube showing osmotic pressureOn the left side of the U-tube is an aqueous solution, and on the right side is pure water. The pure water is trying to dilute the solution by travelling through the semipermeable membrane. Eventually the added weight of the extra water on the left causes enough pressure to stop osmosis.

Osmotic pressure is the pressure that needs to be applied to a solution to prevent the inward flow of water across a semipermeable membrane. Osmotic pressure can also be explained as the pressure necessary to nullify osmosis. One way to stop osmosis is to increase the hydrostatic pressure on the solution side of the membrane this ultimately squeezes the solvent molecules closer together, increasing their “escaping tendency.” The escaping tendency of the solution can be raised until it eventually equals that of the molecules in the pure solvent at this point, osmosis will cease. The osmotic pressure is the pressure required to achieve osmotic equilibrium.

Osmotic pressureOsmotic pressure is the pressure required to stop osmosis.

The osmotic pressure (II) of an ideal solution can be approximated by the Morse equation:

Here, i is the van ‘t Hoff factor, M is the molarity of the solution, R is the gas constant, and T is the absolute temperature in Kelvin. We can see from this equation that the amount of solute present in the solution will directly affect the osmotic pressure of the system.

## Tonicity

Tonicity describes the amount of solute in a solution. The measure of the tonicity of a solution, or the total amount of solutes dissolved in a specific amount of solution, is called its osmolarity. Three terms—hypotonic, isotonic, and hypertonic—are used to relate the osmolarity of a cell to the osmolarity of the extracellular fluid that contains the cells. All three of these terms are a comparison between two different solutions (for example, inside a cell compared to outside the cell).

In a hypotonic solution, such as tap water, the extracellular fluid has a lower concentration of solutes than the fluid inside the cell, and water enters the cell. (In living systems, the point of reference is always the cytoplasm, so the prefix hypo– means that the extracellular fluid has a lower concentration of solutes, or a lower osmolarity, than the cell cytoplasm.) It also means that the extracellular fluid has a higher concentration of water than does the cell. In this situation, water will follow its concentration gradient and enter the cell. This may cause an animal cell to burst, or lyse.

In a hypertonic solution (the prefix hyper– refers to the extracellular fluid having a higher concentration of solutes than the cell’s cytoplasm), the fluid contains less water than the cell does, such as seawater. Because the cell has a lower concentration of solutes, the water will leave the cell. In effect, the solute is drawing the water out of the cell. This may cause an animal cell to shrivel, or crenate.

In an isotonic solution, the extracellular fluid has the same osmolarity as the cell. If the concentration of solutes of the cell matches that of the extracellular fluid, there will be no net movement of water into or out of the cell. The cell will retain its “normal” appearance. Blood cells in hypertonic, isotonic, and hypotonic solutions take on characteristic appearances (Figure 4).

Remember that all three of these terms are comparisons between two solutions (i.e. inside and outside the cell). A solution can’t be hypotonic, that would be like saying that Bob is taller. That doesn’t make sense – you need to say that Bob is taller than Mike. You can say that the solution inside the cell is hypotonic to the solution outside the cell. That also means that the solution outside is hypertonic to the solution inside (just like Mike would be shorter than Bob).

Figure 4 Osmotic pressure changes the shape of red blood cells in hypertonic, isotonic, and hypotonic solutions. (credit: modification of work by Mariana Ruiz Villarreal)

Some organisms, such as plants, fungi, bacteria, and some protists, have cell walls that surround the plasma membrane and prevent cell lysis. The plasma membrane can only expand to the limit of the cell wall, so the cell will not lyse. In fact, the cytoplasm in plants is always slightly hypertonic compared to the cellular environment, and water will always enter the plant cell if water is available. This influx of water produces turgor pressure, which stiffens the cell walls of the plant (Figure 5). In nonwoody plants, turgor pressure supports the plant. If the plant cells become hypertonic, as occurs in drought or if a plant is not watered adequately, water will leave the cell. Plants lose turgor pressure in this condition and wilt.

Figure 5 The turgor pressure within a plant cell depends on the tonicity of the solution that it is bathed in. (credit: modification of work by Mariana Ruiz Villarreal)

But, before entering into the subject of its relationship with the field of health and medicine, we briefly define the general features of what is and what hydrostatic pressure consists of. Let’s see…

### What is hydrostatic pressure?

Hydrostatic pressure refers to the pressure that any fluid in a confined space exerts. If the liquid is in a container, there will be some pressure on the wall of that container.

The hydrostatic pressure is the pressure that is generated by the weight of the liquid on a measurement point, when the liquid is at rest.

The height of a column of liquid, of uniform density, is directly proportional to the hydrostatic pressure.

The hydrostatic properties of a liquid are not constant and the main factors that influence it are the density of the liquid and the local gravity.

It is necessary to know both quantities to determine the hydrostatic pressure of a particular liquid.

Hydrostatic pressure is the force that fluid molecules exert on each other due to the gravitational attraction of the Earth.

This force occurs if the fluid is in motion or at a complete stop, and forces the fluids forward or outward when encountering an area of ​​least resistance in their field.

It is this energy that forces the water from a hole in a paper cup, the gas from a leak in a pipe and the blood from the vessels to the surrounding tissues.

### Increasing the elevation increases the amount of hydrostatic pressure

The fluid that flows downwards also increases the pressure, which causes the water that travels through the waterfalls to flow faster than the water that flows through the stream until it falls.

Temperature is another factor that affects pressure because when temperatures rise, the molecules move at a faster rate, increasing the pressure.

Industries commonly use hydrostatic pressure testing methods to ensure that liquids remain in confined environments.

The tests not only ensure that the pipes and other types of containers do not leak, but also verify that the materials can withstand a greater pressure of possible environmental changes.

It is not uncommon for companies to exert internal forces 150 times more than normal, while controlling pressure changes with instrumentation.

If we imagine a container in the form of a column, we can see that the pressure that pushes against its wall is greater in the background, which will be in the upper part. This is related in part to the force of gravity.

The capillaries are the equivalent of a container in the form of a column, rotated on its side. The pressure that blood exerts on the capillaries is known as blood pressure.

The force of the hydrostatic pressure means that the blood moves along the capillary, the fluid moves through its pores and into the interstitial space.

This movement means that the pressure exerted by the blood will become lower, the blood moves along the capillary, from the arterial to the venous end.

Fluid or hydrostatic statics is the branch of fluid mechanics that studies incompressible fluids at rest.

It covers the study of the conditions under which the fluids are at rest in stable equilibrium against fluid dynamics, the study of fluids in motion.

The hydrostatics are classified as part of the static fluid, which is the study of all fluids, incompressible or not, at rest.

Hydrostatics is fundamental for hydraulics, the engineering of equipment for storing, transporting and using fluids.

It is also relevant to geophysics and astrophysics (for example, in the understanding of plate tectonics and the anomalies of the Earth’s gravitational field), to meteorology, to medicine (in the context of blood pressure) and to many other fields.

Hydrostatics offers physical explanations for many phenomena of everyday life, such as why atmospheric pressure changes with altitude, why wood and oil float on water and why the surface of water is always flat and horizontal, whatever be the shape of your container.

### Pressure in resting liquids

Due to the fundamental nature of the fluids, a fluid can not remain at rest under the presence of a shear stress. However, fluids can exert a normal pressure on any contact surface.

If it is considered that a point of the fluid is an infinitesimally small cube, then it follows from the equilibrium principles that the pressure on each side of this fluid unit must be equal.

If it were not so, the fluid would move in the direction of the resultant force.

Thus, the pressure on a fluid at rest is isotropic That is, it acts with equal magnitude in all directions.

This feature allows fluids to transmit force through the length of pipes or tubes That is, a force applied to a fluid in a pipeline is transmitted, through the fluid, to the other end of the pipe.

This principle was formulated first, in a slightly extended form, by Blaise Pascal, and is now called Pascal’s law.

In a fluid at rest, all frictional and inertial forces disappear and the state of tension of the system is called hydrostatic.

When this condition of V = 0 is applied to the Navier-Stokes equation, the pressure gradient becomes a function of the forces of the body only.

For a barotropic fluid in a field of conservative force as a gravitational force field, the pressure exerted by a fluid in equilibrium becomes a function of the force exerted by gravity.

### Hydrostatic pressure in the field of medicine

Blood vessels have a unique way of maintaining adequate pressure throughout the body. Hydrostatic capillary arterial pressure usually measures 35 millimeters of mercury, or 35 mm Hg. Venous capillary pressure typically measures 15 mm Hg.

The force behind the contractions of the heart along with the gravity that pulls the blood away from the heart causes increasing pressure.

The porous nature of the venous capillaries also decreases the pressure of the flowing blood.

The liquid components of the blood flow naturally through the pores into the interstitial tissues due to this pressure, leaving behind lipids, proteins and particles too large to escape.

This usually decreases venous pressure. On the contrary, the pressure increases within the tissues exerts force towards the capillaries, which is called hydrostatic osmotic pressure .

While the osmotic pressure pushes fluids into the capillary pores, the electrical charges of the solids inside the vessel cause the molecules to bind as they flow into the blood.

This reaction is called the Gibbs-Donnan effect.

The osmotic pressure and the Gibbs-Donnan effect working together, extract fluids from the interstitial tissues into the plasma, which is known as colloid osmotic pressure.

When the body perceives an abnormally low amount of venous pressure, the arteries usually compensate for narrowing.

When damage occurs in the vessel, the plasma contains an insufficient number of solids, or lowers blood pressure, then edema or swelling occurs.

#### Capillary hydrostatic pressure:

This pressure drives fluid out of the capillary (ie, filtration), and is higher at the arteriolar end of the capillary and lower at the venular end.

Depending on the organ, the pressure can fall along the capillary at 15-30 mmHg (axial or longitudinal pressure gradient).

The axial gradient favors filtration at the arteriolar end and reabsorption at the venular end of the capillary.

#### Tissue pressure (interstitial):

This hydrostatic pressure is determined by the volume of interstitial fluid and the compliance of the tissue interstitium, which is defined as the change in volume divided by the change in pressure.

The more fluid that seeps into the gap, the greater the volume of the interstitial space and the hydrostatic pressure within that space. In some organs, interstitial compliance is low, which means that small increases in interstitial volume lead to large increases in pressure.

Examples of this include the brain and the kidney, which are coated by rigid bone (brain) or by a capsule (kidney).

Conversely, soft tissues such as skin, muscles and lungs have a high compliance and, therefore, the interstitial space can undergo a large expansion with a relatively small increase in pressure.

As the interstitial volume increases, the interstitial pressure increases, which may limit the amount of leakage in the interstitium because this pressure opposes capillary hydrostatic pressure.

In other words, as the hydrostatic pressure gradient decreases due to the increase in interstitial pressure, the fluid filtration will be attenuated. However, large increases in tissue interstitial pressure can lead to tissue damage and cell death.

Normally, the interstitial pressure is close to zero. In some tissues it is slightly subatmospheric, while in others it is slightly positive.

#### Capillary capillary oncotic pressure:

Because the capillary barrier is easily permeable to ions, the osmotic pressure within the capillary is mainly determined by plasma proteins that are relatively impermeable.

Therefore, instead of talking about “osmotic” pressure, this pressure is called “oncotic” or “colloidal osmotic” pressure because it is generated by colloids.

Albumin generates approximately 70% of the oncotic pressure. This pressure is typically 25-30 mmHg.

Oncotic pressure increases throughout the capillary, particularly in capillaries that have a high net filtration (for example, in renal glomerular capillaries), because the filtering fluid leaves proteins that lead to an increase in protein concentration.

Normally, when oncotic pressure is measured, it is measured through a semipermeable membrane that is permeable to fluids and electrolytes, but not to large protein molecules.

In most capillaries, however, the wall (mainly endothelium) has a finite permeability to proteins.

The actual permeability to the protein depends on the type of capillarity as well as on the nature of the protein (size, shape, charge).

Due to this finite permeability, the actual oncotic pressure generated through the capillary membrane is less than that calculated from the protein concentration.

The effects of finite protein permeability on physiological oncotic pressure can be determined by knowing the reflection coefficient (σ) of the capillary wall.

If the capillary is impermeable to the protein then it is equal to 1.

When the value for σ is very low, the oncotic pressures of the plasma and tissue may have a negligible influence on the net driving force.

#### Wound (interstitial):

The oncotic pressure of the interstitial fluid depends on the interstitial protein concentration and the reflection coefficient of the capillary wall.

The more permeable the capillary barrier to proteins, the greater the interstitial oncotic pressure.

This pressure is also determined by the amount of fluid filtration in the gap. For example, increased capillary filtration decreases interstitial protein concentration and reduces oncotic pressure.

A reduction in interstitial oncotic pressure increases the net oncotic pressure through the capillary endothelium, which opposes filtration and promotes reabsorption thus serving as a mechanism to limit capillary leakage.

In a “typical” tissue, the oncotic pressure of the tissue is about 5 mmHg (that is, much lower than the oncotic pressure of the capillary plasma).

### What is the difference between oncotic and hydrostatic pressure?

Hydrostatic pressure increases filtration by pushing the fluid and solute out of the capillaries, while capillary oncotic pressure (also known as colloid osmotic pressure) draws fluid into the capillaries and / or prevents hydrostatic pressure.

The hydrostatic pressure is based on the pressure exerted by the pressure of the blood against the walls of the capillaries, while the oncotic pressure exists due to proteins, such as albumin, globulins and fibrinogen, which do not leave the capillary and extract water.

The same forces also act on the interstitial fluid.

The arteries transport oxygenated blood and nutrients to the body’s metabolic tissues. This oxygenated blood travels through the capillary network within the tissues.

The exchange of fluids in the blood capillaries is called microcirculation. The hydrostatic and oncotic pressure are the two types of driving forces that intervene in the movement of fluids during microcirculation.

The main difference between hydrostatic and oncotic pressure is that hydrostatic pressure is the force that pushes the fluid out of the blood capillaries, while the oncotic pressure is the force that pushes the fluid into the blood capillaries.

The general interaction between hydrostatic pressure and oncotic pressure is described by the Starling principle .

## What is Hydrostatic Pressure

Hydrostatic pressure is the pressure at any point of a non-flowing liquid due to the force of gravity. Consider a jar of water. The pressure at the surface of the water is atmospheric pressure. That is the pressure applied on the water by the atmosphere. But if we consider a point at the middle of the water in that jar, the pressure in that point is different from that of the surface. That is because the water above that point also applies a pressure on that point due to gravity.

Figure 1: A jar or container showing the density (d) of the water and the depth to the middle point of the jar

The above image shows a jar of water. There are three points marked on it. The pressure at the point on the surface of the water is atmospheric pressure. This atmospheric pressure can be given as π. The point in the middle is at a depth of h from the surface. The pressure applied by a liquid is given as

P is the pressure applied

h is the depth or the height of the liquid body

d is the density of the liquid

Therefore, the pressure on the middle point in above image can be given as,

The pressure on the bottom of the jar is,

Therefore, the hydrostatic pressure at different points of the same liquid is different. But the hydrostatic pressure at points in the same level of the same liquid is the same.

Figure 2: A jar of water showing three points located at the same level.

In the above image, “a”, “b”, and “c” are located on the same level. Therefore, the pressure at each point would be same. This hydrostatic pressure causes different velocities in water flow at different points of the same liquid. This phenomenon is shown in the below diagram.

Figure 3: Different velocities of water at different levels.

In the above image, A, B and C are holes located at different levels in the same jar of water. The highest velocity is observed at point C. This is because a higher pressure is applied to the point C. The lowest velocity is observed at A since only the atmospheric pressure is applied to that point.

## Osmosis

If two solutions of different concentration are separated by a semi-permeable membrane which is permeable to to the smaller solvent molecules but not to the larger solute molecules, then the solvent will tend to diffuse across the membrane from the less concentrated to the more concentrated solution. This process is called osmosis.

Osmosis is of great importance in biological processes where the solvent is water. The transport of water and other molecules across biological membranes is essential to many processes in living organisms. The energy which drives the process is usually discussed in terms of osmotic pressure.

## Methods

The bi-layer system as well as a typical concentration profile (in the conventional mode) is shown in Fig. 1.

The solute and solvent transport equations in the barrier layer read this way

where c is the reference (virtual) solute concentration (Reference (virtual) solution is defined as such that could be in thermodynamic equilibrium with a given point inside the membrane see ref. 18 for more detail), ω is the solute permeability, σ is the solute reflection coefficient, Js is the solute flux, Jv is the volume flux, is the mechanical permeability, P is hydrostatic pressure, π is osmotic pressure (both in the reference solution). In the limiting case of σ = 1, equations (6 and 7) reduce to the solution-diffusion model 19 .

If the material constants are independent of either coordinate or concentration (Spiegler-Kedem model 16 ), equation (7) can be integrated over the barrier-layer thickness

is the hydraulic permeance of the barrier layer (as customary, we neglect the hydraulic resistance of the porous support).

At constant material constants, equation (6) can be integrated, too

L is the barrier-layer thickness, cd is the solute concentration in the draw solution, ci is the solute concentration at the barrier-layer/support interface. In the support

where De is the effective diffusion coefficient of the solute (accounting for the finite porosity and pore tortuosity). Equation (12) can also be easily integrated.

δ is the support thickness, cf is the solute concentration in the more dilute (feed) solution.

Due to the one-dimensionality of the fluxes, both solute flux, Js, and volume flux, Jv, are the same in the barrier layer and the support. Therefore, from equation (10) and equation (13), we obtain

For simplicity, we neglect the external concentration polarization (existence of unstirred solution layers close to the membrane surfaces). In this case, the draw- and feed-side concentrations at the membrane surfaces, cd and cf, are known. The Péclet numbers Pem and Pes are directly proportional to the transmembrane volume flow, which is given by equation (8).

In the case of non-retarded osmosis (ΔP = 0), by additionally assuming the solution to be ideal, we can relate the trans-membrane volume flow to the concentration difference across the barrier layer.

where v is the salt stoichiometric coefficient. By using the definitions of Péclet numbers of equation (11) and equation (14) after some identical transformations we obtain this transcendental equation for the dimensionless concentration difference across the barrier layer,

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