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Limitations to the Arrhenius Theory
The Arrhenius theory has many more limitations than the other two theories. The theory suggests that in order for a substance to release either H+ or OH- ions, it must contain that particular ion. However, this does not explain the weak base ammonia (NH3) which, in the presence of water, releases hydroxide ions into solution, but does not contain OH- itself.
Hydrochloric acid is neutralized by both sodium hydroxide solution and ammonia solution. In both cases, you get a colourless solution which you can crystallize to get a white salt - either sodium chloride or ammonium chloride. These are clearly very similar reactions. The full equations are:

Strong and Weak Acids and Bases
Strong acids are molecular compounds that essentially ionize to completion in aqueous solution, disassociating into H+ ions and the additional anion; there are very few common strong acids. All other acids are "weak acids" that incompletely ionized in aqueous solution. Acids and bases that dissociate completely are said to be strong acids, e.g.:

Limitations to the Arrhenius Theory
The Arrhenius theory has many more limitations than the other two theories. The theory suggests that in order for a substance to release either H+ or OH- ions, it must contain that particular ion. However, this does not explain the weak base ammonia (NH3) which, in the presence of water, releases hydroxide ions into solution, but does not contain OH- itself.
Hydrochloric acid is neutralized by both sodium hydroxide solution and ammonia solution. In both cases, you get a colourless solution which you can crystallize to get a white salt - either sodium chloride or ammonium chloride. These are clearly very similar reactions. The full equations are:

Strong and Weak Acids and Bases
Strong acids are molecular compounds that essentially ionize to completion in aqueous solution, disassociating into H+ ions and the additional anion; there are very few common strong acids. All other acids are "weak acids" that incompletely ionized in aqueous solution. Acids and bases that dissociate completely are said to be strong acids, e.g.:

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Here the right-handed arrow (→→) implies that the reaction goes to completion. That is, a 1.0 M solution of HClO4 in water actually contains 1.0 M H+(aq) and 1.0 M ClO4-(aq), and no undissociated HClO4.Conversely, weak acids such as acetic acid (CH3COOH) and weak bases such as ammonia (NH3) dissociate only slightly in water - typically a few percent, depending on their concentration and exist mostly as the undissociated molecules.STRONG ACIDS: HCl, HNO3, H2SO4, HBr, HI, HClO4
WEAK ACIDS: All other acids, such as HCN, HF, H2S, HCOOH
Strong acids such as HClHCl dissociate to produce spectator ions such as Cl−Cl− as conjugate bases, whereas weak acids produce weak conjugate bases. This is illustrated below for acetic acid and its conjugate base, the acetate anion. Acetic acid is a weak acid (Ka = 1.8 x 10-5) and acetate is a weak base (Kb = Kw/Ka = 5.6 x 10-10)

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Here the right-handed arrow (→→) implies that the reaction goes to completion. That is, a 1.0 M solution of HClO4 in water actually contains 1.0 M H+(aq) and 1.0 M ClO4-(aq), and no undissociated HClO4.Conversely, weak acids such as acetic acid (CH3COOH) and weak bases such as ammonia (NH3) dissociate only slightly in water - typically a few percent, depending on their concentration and exist mostly as the undissociated molecules.STRONG ACIDS: HCl, HNO3, H2SO4, HBr, HI, HClO4
WEAK ACIDS: All other acids, such as HCN, HF, H2S, HCOOH
Strong acids such as HClHCl dissociate to produce spectator ions such as Cl−Cl− as conjugate bases, whereas weak acids produce weak conjugate bases. This is illustrated below for acetic acid and its conjugate base, the acetate anion. Acetic acid is a weak acid (Ka = 1.8 x 10-5) and acetate is a weak base (Kb = Kw/Ka = 5.6 x 10-10)

Like acids, strong and weak bases are classified by the extent of their ionization. Strong bases disassociate almost or entirely to completion in aqueous solution. Similar to strong acids, there are very few common strong bases. Weak bases are molecular compounds where the ionization is not complete.
STRONG BASES: The hydroxides of the Group I and Group II metals such as LiOH, NaOH, KOH, RbOH, CsOH
WEAK BASES: All other bases, such as NH3, CH3NH2, C5H5N

pH Scale
Since acids increase the amount of H+ ions present and bases increase the amount of OH- ions, under the pH scale, the strength of acidity and basicity can be measured by its concentration of H+ ions. This scale is shown by the following formula:
pH = -log[H+]
with [H+] being the concentration of H+ ions.
To see how these calculations are done, refer to Calculating the pH of the solution of a Polyprotic Base/Acid
The pH scale is often measured on a 1 to 14 range, but this is incorrect (see pH for more details). Something with a pH less than 7 indicates acidic properties and greater than 7 indicates basic properties. A pH at exactly 7 is neutral. The higher the [H+], the lower the pH.

Lewis Theory
The Lewis theory of acids and bases states that acids act as electron pair acceptors and bases act as electron pair doners. This definition doesn't mention anything about the hydrogen atom at all, unlike the other definitions. It only talks about the transfer of electron pairs. To demonstrate this theory, consider the following example.

This is a reaction between ammonia (NH3) and boron trifluoride (BF3). Since there is no transfer of hydrogen atoms here, it is clear that this is a Lewis acid-base reaction. In this reaction, NH3 has a lone pair of electrons and BF3 has an incomplete octet, since boron doesn't have enough electrons around it to form an octet.