To avoid concluding that the value of x is always exactly half the value of c (i.e., x = c/2) in the expression x^2/(c-x), we can analyze the situation using a general approach and investigate the conditions under which this conclusion may or may not hold.
Let's start by setting up the equation for the equilibrium constant expression:
Ka = [A-][H+]/[HA]
Where [A-] represents the concentration of the conjugate base, [H+] represents the concentration of the hydronium ion, and [HA] represents the concentration of the acid. In this case, we are given that Ka = 5*10^-1.
Now, let's consider the acid dissociation reaction:
HA ⇌ A- + H+
Let's assume that the initial concentration of HA is denoted as c. At equilibrium, let's assume the concentration of A- is x, and the concentration of H+ is also x. Therefore, the concentration of HA at equilibrium is c - x.
Now, using the equilibrium constant expression, we have:
Ka = x * x / (c - x)
We can rearrange the equation as follows:
x^2 = Ka * (c - x)
Expanding the equation:
x^2 = Ka * c - Ka * x
Rearranging and collecting terms:
x^2 + Ka * x - Ka * c = 0
Now, we have a quadratic equation in terms of x. To solve this equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, the coefficients are:
a = 1 b = Ka c = -Ka * c
Substituting these values into the quadratic formula, we get:
x = (-Ka ± √(Ka^2 + 4Ka * c)) / 2
From this equation, we can see that the value of x is dependent on both Ka and the initial concentration of HA (c). Therefore, it is not always the case that x is exactly half the value of c. The specific value of x will depend on the specific values of Ka and c.
So, it is incorrect to conclude that x/c always equals 0.5 or 50% at any dilution. The value of x/c will depend on the specific equilibrium conditions and concentrations involved.