for solutions of a weak acid, a graph of pH versus the logarithm of the initial acid concentration should be a straight line. What is the magnitude of the slope of that line?
http://www.rpi.edu/dept/chem-eng/Biotech-Environ/EQUILIBRIUM/logconc.htm
Click on the link at the bottom "explanation of why the lines are linear".
To understand the magnitude of the slope in a graph of pH versus the logarithm of the initial acid concentration for a weak acid solution, let's first discuss some background information.
The pH of a solution is a measure of its acidity or basicity and is defined as the negative logarithm (base 10) of the hydrogen ion concentration ([H+]) in the solution. pH = -log[H+].
For a weak acid (HA), it undergoes dissociation in water to produce H+ and its conjugate base A-. The equilibrium constant expression for the dissociation reaction is written as follows:
HA ⇌ H+ + A-
The equilibrium constant for this reaction is denoted as Ka and is defined as the ratio of the concentration of the products (H+ and A-) to the concentration of the reactant (HA) at equilibrium.
Ka = [H+][A-] / [HA]
To express the ionization of the weak acid in terms of the degree of dissociation (α), the following equation can be used:
[H+] = α[HA]₀
Where [HA]₀ is the initial concentration of the weak acid.
Taking the negative logarithm of both sides of the equation, we get:
-pH = log(α[HA]₀)
Since the concentration of [HA]₀ can vary, it is more convenient to work with the logarithm of this term. Therefore, we take the logarithm of the equation:
-log(α[HA]₀) = log(α) + log([HA]₀)
Using the definition of pH (-log[H+]), we can rewrite the equation as:
-pH = log(α) + log([HA]₀)
Comparing this equation with the equation of a straight line (y = mx + c), we can see that the slope of the graph of pH versus log([HA]₀) is equal to the degree of dissociation (α).
Hence, the magnitude of the slope of the graph is equal to the degree of dissociation (α) of the weak acid.