An aqueous solution of NaOH has a concentration of 0.01 mol/L.
Given the ionic product of water is K_w = [H^+] [OH^–] = 10^{–14} mol^2/L^2 (at 25°C) and that the equation for pH is pH = – log_{10} [H^+], calculate the pH of the NaOH solution at 25°C.
A. 13
B. 11
C. 7
D. 14
E. 12
To solve this problem, you can use the ionic product of water and the equation for pH to calculate the concentration of hydrogen ions in the NaOH solution. Since the ionic product of water is given as K_w = [H^+] [OH^–] = 10^{–14} mol^2/L^2, and the concentration of hydroxide ions in the NaOH solution is known to be 0.01 mol/L, you can rearrange the ionic product equation to solve for the concentration of hydrogen ions in the solution:
[H^+] [OH^–] = K_w
[H^+] = K_w / [OH^–]
[H^+] = 10^{–14} mol^2/L^2 / 0.01 mol/L
[H^+] = 10^{–12} mol/L
Next, you can use the equation for pH to calculate the pH of the solution. The equation for pH is given as pH = – log_{10} [H^+], so you can plug in the value of [H^+] that you just calculated to solve for the pH of the solution:
pH = – log_{10} [H^+]
pH = – log_{10} (10^{–12} mol/L)
pH = – (–12)
pH = 12
Therefore, the pH of the NaOH solution at 25°C is 12, which means that it is a basic solution. The correct answer to the question is option E, 12.
o solve this problem without a calculator, you can use a technique called “scientific notation” to quickly and easily perform the calculations that are required. Scientific notation is a way of expressing very large or very small numbers in a more compact and manageable form. In this case, you can use scientific notation to simplify the calculations that are required to solve the problem.
For example, instead of using the actual value of the ionic product of water, which is K_w = 10^{–14} mol^2/L^2, you can use a simplified version of this value that is easier to work with. For example, you could use the value K_w = 1 * 10^{–14} mol^2/L^2, which is equivalent to the original value, but it is easier to manipulate mathematically.
Next, you can use the value of K_w that you just calculated to solve for the concentration of hydrogen ions in the NaOH solution. Since the concentration of hydroxide ions in the solution is given as 0.01 mol/L, you can use this value to solve for the concentration of hydrogen ions in the solution:
[H^+] [OH^–] = K_w
[H^+] = K_w / [OH^–]
[H^+] = 1 * 10^{–14} mol^2/L^2 / 0.01 mol/L
[H^+] = 10^{–12} mol/L
Finally, you can use the equation for pH to calculate the pH of the solution. The equation for pH is given as $$pH = – log_{10} [H^+]$$, so you can plug in the value of [H^+] that you just calculated to solve for the pH of the solution:
pH = – log_{10} [H^+]
pH = – log_{10} (10^{–12} mol/L)
pH = – (–12)
pH = 12
Therefore, the pH of the NaOH solution at 25°C is 12, which means that it is a basic solution. The correct answer to the question is option E, 12.
By using scientific notation and a few simple mathematical manipulations, you can solve this problem without the need for a calculator. This technique can be useful for quickly solving pH questions and other problems that involve very large or very small numbers.
If you are not familiar with the math above, you can also count the number if 0 in the H+ concentration, or OH-, to find the pH, or pOH.