Lever rule derivation

In an undoped, intrinsic semiconductor the equiHbrium concentrations of electrons, and holes,/), are described by a lever rule derived from the law of mass action (eq. 3) ... 

The lever rule can be derived from the mass balance as well as from the balance of the amount of substance. On a straight line we can show a mixture X lying in between two pure components A and B and their respective mole fractions and the amount of substances ... 

Graphical solutions to material balance problems involving equilibrium relationships offer the advantages of speed and convenience. Fundamental to all graphical methods is the so-called inverse lever rule, which is derived in Example 3.3 and applied in Example 3.4. 

Since the derivation of the lever rule depends only on a mass balance, the rule is valid for calculating the relative amounts of the two phases present in any two-phase region of a two-component system. If the diagram is drawn in terms of mass fraction instead of mole fraction, the level rule is valid and yields the relative masses of the two phases rather than the relative mole numbers. 

Solutions of (6.14) and (6.15), the rectifying and stripping cascade flash trajectories, can be represented in mole fraction space (three dimensional for the IPOAc system). However, we represent the solutions in transformed composition space, which is two dimensional for IPOAc system (for a derivation and properties of these transformed variables [46]). This transformed composition space is a projection of a three dimension mole fraction space onto a two dimensional transformed composition subspace for the IPOAc system. Even though the correspondence between real compositions and transformed compositions is not one-to-one in the kinetic regime, we will make use of these transforms because of ease of visualization of the trajectories, and because overall mass balance for reactive systems (kinetically or equilibrium limited) can be represented with a lever rule in transformed compositions. We use this property to assess feasible splits for continuous RD. 

This result states that if two forces with magnitude Xl and Xv act on points L and V, respectively, their torques with respect to point E are equal. This property gives the name lever rule to eq. f2.iol. The lever analogy has be used as a mnemonic trick to memorize the equation. No memorization is required, however eq. r2.io ) can be derived easily when needed by applying a straightforward mass balance. 

We can derive a more general form of the lever rule that will be needed in Chap. 13 for phase diagrams of multicomponent systems. This general form can be applied to any two-phase area of a two-dimensional phase diagram in which a tie-line construction is valid, with the position of the system point along the tie line given by the variable... 

The lever rule works, according to the general derivation in Sec. 8.2.4, because the ratio wa/w, which is equal to ZA, varies linearly with the position of the system point along a tie line on the triangular phase diagram. 

Calculate the flow rate of stream 3 (in mol/min) in terms of a and p. If you derived this correctly you proved the lever rule (see Chapter 4). You are not permitted to invoke the lever rule to calculate the flow rate of stream 3. 

We now establish how much of each phase must be present to form a heterogeneous mixture of average mole fraction x. You should consult Table 3.8.1, in deriving the so-called lever rule. 

Thus, the lever rule may be employed to determine the relative amounts or fractions of phases in any two-phase region for a binary alloy if the temperature and composition are known and if equilibrium has been established. Its derivation is presented as an example problem. 

The existence of these straight lines and the applicability of the lever-arm rule can be proved by deriving Eq. (13z44). 

His text might strike the reader as bizarre there are no equations his analysis is throughout in terms of ratio and proportion. In contrast to the box of the scaling rules for beams taken from the Mechanics Text-Book published two centuries down the road, Galileo provides a derivation of the resistance of a cantilever to fracture due to an end load. His analysis relies on the principle of equilibrium of the lever his result expresses the fracture load in terms of the ratio of the length of the beam to its thickness and a property of the material - its resistance when subject to tension. 

---