Volume flux density

When the water potential inside a cell differs from that outside, water is no longer in equilibrium and we can expect a net water movement from the region of higher water potential toward the region of lower water potential. This volume flux density of water, Jyw> is generally proportional to the difference in water potential (AY) across the membrane or membranes restricting the flow. The proportionality factor indicating the permeability to water flow at the cellular level is expressed by a water conductivity coefficient, L,1 ... 

When we incorporate this last relation into our expression for the volume flux density of water, we obtain... 

Internodal cells of Cham and Nitella grow relatively slowly — a change in volume of about 1% per day is a possible growth rate for the fairly mature cells used in determining Am- This growth rate means that the water content increases by about 1% of the volume per day (1 day = 8.64 x 104 s). The volume flux density for water is the rate of volume increase divided by the surface area across which water enters ... 

A. What is the initial net volume flux density of water into or out of the cell when it is placed in pure water at atmospheric pressure ... 

In the next section we will use Equation 3.29 as the starting point for developing the expression from irreversible thermodynamics that describes the volume flux density. Because the development is lengthy and the details may obscure the final objective, namely, the derivation of Equation 3.40 for the volume flux density in terms of convenient parameters, the steps and the equations involved are summarized first ... 

Now that we have appropriately expressed the chemical potential differences of water and the solute, we direct our attention to the fluxes. Expressed in our usual units, the flux densities Jw and Js are the moles of water and of solute, respectively, moving across 1 m2 of membrane surface in a second. A quantity of considerable interest is the volume flux density Jv, which is the rate of movement of the total volume of both water and solute across unit area of the membrane Jv has the units of volume per unit area per unit time (e.g., m3 m-2 s-1, or m s-1). 

The molar flux density of species j (Jj) in mol m-2 s-1 multiplied by the volume occupied by each mole of species j (V)) in m3 mol-1 gives the volume flux density for that component (iy) in m s-1. Hence, the total volume flux density is... 

It is generally simpler and more convenient to measure the total volume flux density (such as that given in Eq. 3.34) than one of the component volume flux densities (Jv = VjJj). For instance, we can often determine the volumes of cells or organelles under different conditions and relate any changes in volume to Jy. [The volume flux density of water JVw used in Chapter 2 (e.g., Eq. 2.26) is VWJW.]... 

LP is the hydraulic conductivity coefficient and can have units of m s-1 Pa-1. It describes the mechanical filtration capacity of a membrane or other barrier namely, when An is zero, LP relates the total volume flux density, Jv, to the hydrostatic pressure difference, AP. When AP is zero, Equation 3.37 indicates that a difference in osmotic pressure leads to a diffusional flow characterized by the coefficient Lo Membranes also generally exhibit a property called ultrafiltration, whereby they offer different resistances to the passage of the solute and water.14 For instance, in the absence of an osmotic pressure difference (An = 0), Equation 3.37 indicates a diffusional flux density equal to LopkP. Based on Equation 3.35, vs is then... 

Using the definition of a (Eq. 3.38), we can rewrite Equation 3.36 to obtain the following form for the total volume flux density ... 

Many different solutes can cross a membrane under usual conditions. Each such species j can be characterized by its own reflection coefficient, ay, for that particular membrane. The volume flux density given by Equation 3.39 can then be generalized to... 

Let us consider the realistic situation of AP equaling zero across a membrane bathed on either side by aqueous solutions (e.g., Chapter 2, Section 2.3A). By Equation 3.39 [Jv = LP (AP - oAFI)], the volume flux density (Jv) is then equal to —LPcrATl. For a solute having a reflection coefficient equal to zero for that particular membrane, the volume flux density is zero. By Equation 3.34b (Jv = VWJW +VSJS), a zero Jvimplies that VWJW equals —VSJS. In words, the volume flux density of water must be equal and opposite to the volume flux density of the solute to result in no net volume flux density. Conversely, the absence of a net volume flux density... 

We can profitably reexamine certain aspects of the movement of solutes into and out of cells and organelles by using the more general equations from irreversible thermodynamics. One particularly important situation amenable to relatively uncomplicated analysis occurs when the total volume flux density Jv is zero, an example of a stationary state. This stationary state, in which the volume of the cell or organelle does not change over the time period of interest, can be brought about by having the net volume flux... 

Figure 3-21. Changes in the volume flux density (/y) across a roo t or in the reciprocal of the wa ter volume in an organelle [1/(V - 6)] as the osmotic pressure of a specific solute in the external bathing solution n° is increased. An increase in J7° for a nonzero reflection coefficient of solute species x (trx > 0) decreases the influx or increases the efflux from the root Jy = LP(Ap -Eq. 3.40] or decreases the volume for the organelle [crxn° + a=J Eq. 3.45],...

Measurements of incipient plasmolysis can be made for zero volume flux density (Jv = 0) and for a simple external solution (x° = 0) at atmospheric pressure (P° = 0). In this case, Equation 3.41 is the appropriate expression from irreversible thermodynamics, instead of the less realistic condition of water equilibrium that we used previously. For this stationary state condition, the following expression describes incipient plasmolysis (P1 = 0) when the solutes can cross the cell membrane ... 

Equation 3.48 indicates that not only does Js depend on An, as expected from classical thermodynamics, but also that the solute flux density can be affected by the overall volume flux density, Jv. In particular, the classical expression for Js for a neutral solute is P Ac (Eq. 1.8), which equals (Pj/RT)ATlj using the Van t Hoff relation (Eq. 2.10 II, = PT Cj). Thus to, is analogous to P/RT of the classical thermodynamic description (Fig. 3-19). The classical treatment indicates that Js is zero if An is zero. On the other hand, when An is zero, Equation 3.48 indicates that Js is then equal to c,(l - cr,)/y solute molecules are thus dragged across the membrane by the moving solvent, leading to a solute flux density proportional to the local solute concentration and to the deviation of the reflection coefficient from 1. Hence, Pj may not always be an adequate parameter by which to describe the flux of species , because the interdependence of forces and fluxes introduced by irreversible thermodynamics indicates that water and solute flow can interact with respect to solute movement across membranes. 

C. Suppose that some treatment makes the membrane nonselective for all solutes present. What is the net volume flux density then ... 

Figure 9-12. Cylindrically symmetric flow of soil water toward a root (flow arrows indicated only for outermost cylinder). The volume flux density, Jv, at the surface of each concentric cylinder times the cylinder surface area (2jrr x /) is constant in the steady state, so Jv then depends inversely on r, the radial distance from the root axis.

The uptake of water by a young root 1 mm in diameter (r = 0.5 x 10-3 m) is usually 1 x 1CT5 to 5 x 10-5m3 day-1 per meter of root length (0.1-0.5 cm3 day-1 per centimeter of length). This uptake occurs over a root surface area of 2nd, so the volume flux density of water at the root surface for a moderate water uptake rate of 3 x 10-5 m3 day-1 per meter of root length is... 

Thus the hydrostatic pressure decreases by 0.2 MTa across a distance of 10 mm in the soil next to the root. Assuming that the solute content of the soil water does not change appreciably over this interval, n 011 is 0.1 MlPa (the same as IT 011), and thus T 011 adjacent to the root is -0.4M[Pa - 0.1 M[Pa, or -0.5 MIPa (see Table 9-3). As the soil dries, LSDl1 decreases therefore, the decreases in hydrostatic pressure and in water potential adjacent to a root must then become larger to maintain a given volume flux density of solution toward a root. 

For spherical symmetry we note that Jv4m2 is constant in the steady state, where Am2 is the area of a sphere. As concentric spherical shells of water thus move toward the sphere, the flux density increases inversely as r2. We thus obtain the following steady-state relation describing the volume flux density Jv at distance r from the center of a sphere when Jv varies only in the radial direction and LSDl1 is constant ... 

A steady state is often not achieved in soils, so a steady rate is sometimes used, where the rate of volumetric water depletion is constant, leading to relations considerably more complicated than are Equations 9.8 and 9.9.3 We note that the equations describing water flow in a soil for the onedimensional case (Eq. 9.7), for cylindrical symmetry (Eq. 9.8), and for spherical symmetry (Eq. 9.9) all indicate that the volume flux density of water is proportional to LSGl1 times a difference in hydrostatic pressure (see Table 9-2). 

During germination the volume flux density of water into a seed is often limited by a seed coat (testa) of thickness 8SC (Fig. 9-13). The seed coat is a complex, multilayered, hard, rather impervious tissue that is relatively thin compared to the radius of the seed. For the volume flux density at the seed surface (r = rs> ra = rs> and t b = rs - Ssc Fig. 9-13), Equation 9.9 becomes... 

Next, we will use Poiseuille s law to estimate the pressure gradient necessary to cause a specified volume flux density in the conducting cells of the xylem.4 The speed of sap ascent in the xylem of a transpiring tree can be about 1 mm s-1, which is 3.6 m hour-1. We note that vw equals the volume flux density of water, Jyw (see Chapter 2, Section 2.4F) the average speed of the solution equals JVy the volume flux density of the solution, which for a dilute aqueous solution such as occurs in the xylem is about the same as Jvw (see Chapter 3, Section 3.5C). Thus Jvin the xylem of a transpiring tree can be 1 mm s-1. For a viscosity of 1.0 x 10-3 Pa s and a xylem element with a lumen radius of 20 pm, the pressure gradient required to satisfy Equation 9.11b then is... 

Figure 9-15. Volume flux densities and pressure gradients in cylindrical tubes. The tube radius is 20 [.un, and dP/dx is calculated from Poiseuille s law (Eq. 9.11b) as modified by gravitational effects. For vertical tubes, x is considered positive upward. Arrows indicate the direction of flow.

Let us designate the average volume flux density of water across area A3 of component j by 7y, which is the average velocity of the water movement (Chapter 2, Section 2.4F). A7 can be the root surface area, the effective cross-sectional area of the xylem, or the area of one side of the leaves. In the steady state, the product J v A7 is essentially constant, because nearly all of the water taken up by the root is lost by transpiration that is, the same volume of water moves across each component along the pathway per unit time. We will represent the drop in water potential across component j by AT7 defining the resistance of component j ( ) as follows ... 

Table 9-4. Values for the Volume Flux Density of Water and Relative Areas for Its Flow along a Plant, Illustrating the Constancy of Jjv (Eq. 9.12)...

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