What is the pH of a 1.0E-8 M solution of HCl? Please note that you cannot simply add the contribution of H+ ions from the autoproteolysis of water and the contribution from the dissociation of HCl. You will need to use the quadratic equation to solve this problem and report the answer to 4 figures.
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To find the pH of a solution, we need to determine the concentration of H+ ions present. In this case, we have a solution of HCl with a concentration of 1.0E-8 M.
The dissociation of HCl can be represented by the equation:
HCl (aq) → H+ (aq) + Cl- (aq)
Since HCl is a strong acid, it completely ionizes in water. Therefore, the concentration of H+ ions is equal to the concentration of HCl, which is 1.0E-8 M.
To find the pH, we need to take the negative logarithm (base 10) of the H+ concentration. Therefore, the pH can be calculated as:
pH = -log[H+]
pH = -log(1.0E-8)
At this point, we need to use the quadratic equation to solve for the logarithm since the concentration is extremely small.
In order to do so, we need to rewrite the equation in the form of:
ax^2 + bx + c = 0
In this case, let's assign x as -log[H+]. The equation becomes:
x^2 + log(1.0E-8)x - log(1.0E-8) = 0
Now, we can solve this quadratic equation for x using the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
Let's plug in the values:
a = 1
b = log(1.0E-8) = -8
c = -log(1.0E-8) = 8
x = (-(-8) ± √((-8)^2 - 4*1*8))/(2*1)
x = (8 ± √(64 - 32))/(2)
Simplifying further:
x = (8 ± √32)/(2)
To find the pH value, we take the negative sign because pH represents the negative logarithm:
pH = -x
pH = -(8 ± √32)/(2)
Now, we can calculate the pH value by evaluating the expression above using a calculator:
pH ≈ -1.8
Hence, the pH of the 1.0E-8 M solution of HCl is approximately -1.8. Note that the pH scale typically ranges from 0 to 14, so negative pH values are not conventionally used.