The Brønsted-Lowry acid-base theory states that upon reacting an acid with a base, the acid acts as a proton ($H^{+}$) donor, while the base acts as a proton acceptor.
When an acid is added to water, the acid dissociates releasing $H^{+}$ ions into the solution. For example, when hydrogen chloride ($HCl_{(g)}$) is added to an aqueous solution, it dissociates into $H^{+}_{(aq)}$ and $Cl^{-}_{(aq)}$ ions:
$$ HCl_{(g)} + aq \rightarrow H^{+}_{(aq)} + Cl^{-}_{(aq)} $$
Acids can release multiple protons into the solution, equal to the number of hydrogen atoms in the acid.
Aqueous acids can take part in reactions with bases, carbonates and alkalis. In these reactions, the acid is neutralised forming a salt and water. In writing ionic equations it is important to know that:
Aqueous acids react with solid carbonates forming a salt, carbon dioxide and water.
$$ 2HCl_{(aq)} + CaCO_{3(s)} \rightarrow CaCl_{2(aq)} + CO_{2(g)} + H_{2}O_{(l)} \\
2H^{+}_{(aq)} + CaCO_{3(s)} \rightarrow Ca^{2+}_{(aq)} + CO_{2(g)} + H_{2}O_{(l)} $$
If the carbonate is in solution the carbonate dissociates, so the ionic equation can be further simplified.
$$ 2HCl_{(aq)} + Na_{2}CO_{3(aq)} \rightarrow 2NaCl_{(aq)} + CO_{2(g)} + H_{2}O_{(l)} \\
2H^{+}_{(aq)} + CO_{3(aq)}^{~2-} \rightarrow CO_{2(g)} + H_{2}O_{(l)} $$
Aqueous acids react with bases to form a salt and water.
$$ 2HNO_{3(aq)} + MgO_{(s)} \rightarrow Mg(NO_{3})_{2(aq)} + H_{2}O_{(l)} \\
2H^{+}_{(aq)} + MgO_{(s)} \rightarrow Mg_{(aq)}^{~2+} + H_{2}O_{(l)} $$
Aqueous acids react with alkalis to form a salt and water.
$$ H_{2}SO_{4(aq)} + 2KOH_{(aq)} \rightarrow K_{2}SO_{4(aq)} + 2H_{2}O_{(l)} \\
H^{+}_{(aq)} + OH^{-}_{(aq)} \rightarrow H_{2}O_{(l)} $$
Aqueous acids react with metals to form a salt and hydrogen gas.
$$ 2HCl_{(aq)} + Mg_{(s)} \rightarrow MgCl_{2(aq)} + H_{2(g)} \\
2H^{+}_{(aq)} + Mg_{(s)} \rightarrow Mg^{~2+}_{(aq)} + H_{2(g)} $$
An acid is a proton donor while a base is a proton acceptor. In order for an acid to release a proton, a base must be able to accept a proton. A conjugate pair is a set of two species that transform into each other with the loss or gain of a proton.
Below is the equilibrium reaction showing the dissociation of nitrous acid ($HNO_{2}$) in water.
$$ HNO_{2(aq)} + H_{2}O_{(l)} \rightleftharpoons H_{3}O^{+}_{(aq)} + NO_{2(aq)}^{-} $$
In the forwards reaction:
$$ \eqalign{ &HNO_{2(aq)} + &H_{2}O_{(l)} \rightleftharpoons &H_{3}O^{+}_{(aq)} + &NO_{2(aq)}^{-} \\
&\text{acid 1} &\text{base 2} &\text{acid 2} &\text{base 1}
} $$
The $pH$ scale is an indicator of how acidic or basic a substance is. A substance with a $pH$ less than $7$ is acidic while a substance with a $pH$ more than $7$ is basic. The $pH$ scale shows the concentration of $H^{+}_{(aq)}$ ions in the substance. Due to the large range of possible values for $pH$, a logarithmic scale can be used to make the figures more manageable:
$$ pH = -\log_{10} [H^{+}_{(aq)}] \\
[H^{+}_{(aq)}] = 10^{-pH} $$
A small value of $pH$ means a high concentration of $H^{+}_{(aq)}$ ions, while a high value of $pH$ means a low concentration of $H^{+}_{(aq)}$ ions.
The acid dissociation constant ($K_{a}$) is a measure of the strength of an acid in a solution. For the dissociation of the acid $HA_{(aq)}$:
$$ HA_{(aq)} \rightleftharpoons H^{+}_{(aq)} + A^{-}_{(aq)} $$
The equation for the acid dissociation constant is:
$$ K_{a} = \frac{[H^{+}][A^{-}]}{[HA]} $$
A large value of $K_{a}$ indicates a large extent of dissociation therefore a strong acid. A small value of $K_{a}$ indicates a small extent of dissociation therefore a weaker acid.
A logarithmic scale can be used in order to make the values of $K_{a}$ more manageable. The smaller the value of $pK_{a}$, the stronger the acid.
$$ \eqalign{pK_{a} &= -\log_{10} K_{a} \\
K_{a} &= 10^{-pK_{a}}} $$
For an acid-base equilibrium set up in aqueous solution:
$$ HA_{(aq)} \rightleftharpoons H^{+}_{(aq)} + A^{-}_{(aq)} $$
The strength of the acid $HA_{(aq)}$ is determined by the extent of the dissociation of the $H^{+}_{(aq)}$ and $A^{-}_{(aq)}$ ions.
Strong acids dissociate completely in aqueous solutions. These typically have a $pH \leq 1 $. This means that for:
Once the $[H^{+}_{(aq)}]$ has been calculated, the $pH$ can be calculated directly with $pH = -\log_{10} [H^{+}_{(aq)}]$.
In the dissociation of weak acids:
From these assumptions, the acid dissociation equation can be shown as below. This equation can be used to calculate the $[H^{+}]$ from known values of $K_{a}$ and $[HA]$.
$$ K_{a} = \frac{[H^{+}]^{2}}{[HA]} \\
[H^{+}] = \sqrt{K_{a} \times [HA]}$$
Pure water dissociates into $H^{+}_{(aq)}$ and $OH^{-}_{(aq)}$ ions in an endothermic reaction. For the dissociation of water, the equation for $K_{a}$ is:
$$ K_{a} = \frac{[H^{+}_{(aq)}][OH^{-}_{(aq)}]}{[H_{2}O_{(l)}]} $$
As $[H_{2}O_{(l)}]$ and $K_{a}$ are both constants, these two values can be combined to give the ionic product of water, $K_{w}$.
$$ K_{w} = [H^{+}_{(aq)}][OH^{-}_{(aq)}] $$
$K_{w}$ controls the balance between $[H^{+}_{(aq)}]$ and $[OH^{-}_{(aq)}]$ in all aqueous solutions. At $25°$, the value of $K_{w}$ is $1.00 \times 10^{-14}~mol^{2}dm^{-6}$.
A base is a proton acceptor. An alkali is a soluble base that releases hydroxide ions ($OH^{-}$) when dissolved in water. The strength of a base is its ability to dissociate to generate $OH^{-}$ ions.
$$ NaOH_{(aq)} + aq \rightarrow Na^{+}_{(aq)} + OH^{-}_{(aq)} $$
In order to find the $pH$, the $[H^{+}_{(aq)}]$ must be determined. This can be done by using the ionic product of water.
$$ [H^{+}_{(aq)}] = \frac{K_{w}}{[OH^{-}_{(aq)}]} $$
A strong alkali completely dissociates in aqueous solutions. This means that the $[OH^{-}_{(aq)}]$ is equal to the concentration of the base for monobasic bases such as $NaOH$ or $KOH$.