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where P_M is the permeability of the protoplasm to molecules, M_o, denotes the external and M_i the internal concentration of molecules, A_i denotes the internal concentration of the anion A^- and

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(It is assumed that the activity coefficients equal 1.) Putting P_MF_M = V_M, the apparent velocity constant of the process, we have

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where e denotes the concentration at equilibrium. Then

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where t is time.

The corresponding equation when ions alone enter is

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where K is the dissociation constant of HA, P_A is the permeability of the protoplasm to the ion pair H⁺ + A^-, and A_ie denotes the internal concentration of A_i at equilibrium. Putting P_AKF_M = V_A, the apparent velocity constant of the process, we have

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and

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When both ions and molecules of HA enter together we have

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where S_i = M_i + A_i and S_ie is the value of S_i at equilibrium. Then

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V_M, V_A, and V_MA depend on F_M and hence on the internal pH value but are independent of the external pH value except as it affects the internal pH value.

When the ion pair Na⁺ + A^- penetrates and Na_i = BA_i, we have

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and

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where P_NaA is the permeability of the protoplasm to the ion pair Na⁺ + A^-, Na_o and Na_i are the external and internal concentrations of Na⁺,

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and V_Na is the apparent velocity constant of the process.

Equations are also given for the penetration of:

(1) molecules of HA and the ion pair Na⁺ + A^-,

(2) the ion pairs H⁺ + A^- and Na⁺ + A^-,

(3) molecules of HA and the ion pairs Na⁺ + A^- and H⁺ + A^-.

(4) The penetration of molecules of HA together with those of a weak base ZOH.

(5) Exchange of ions of the same sign.

When a weak electrolyte HA is the only solute present we cannot decide whether molecules alone or molecules and ions enter by comparing the velocity constants at different pH values, since in both cases they will behave alike, remaining constant if F_M is constant and falling off with increase of external pH value if F_M falls off. But if a salt (e.g., NaA) is the only substance penetrating the velocity constant will increase with increase of external pH value: if molecules of HA and the ions of a salt NaA. penetrate together the velocity constant may increase or decrease while the internal pH value rises.

The initial rate

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(i.e., the rate when M_i = 0 and A_i = 0) falls off with increase of external pH value if HA alone is present and penetrates as molecules or as ions (or in both forms). But if a salt (e.g., NaA) penetrates the initial rate may in some cases decrease and then increase as the external pH value increases.

At equilibrium the value of M_i equals that of M_o (no matter whether molecules alone penetrate, or ions alone, or both together). If the total external concentration (S_o = M_o + A_o) be kept constant a decrease in the external pH value will increase the value of M_o and make a corresponding increase in the rate of entrance and in the value at equilibrium no matter whether molecules alone penetrate, or ions alone, or both together.

What is here said of weak acids holds with suitable modifications for weak bases and for amphoteric electrolytes and may also be applied to strong electrolytes.