PH Calculation: 0.273 M Acetic Acid Solution

by Alex Johnson 45 views

Calculating the pH of a weak acid solution, like acetic acid, involves understanding equilibrium concepts and using the acid dissociation constant (Ka). In this comprehensive guide, we'll walk through the process step-by-step, ensuring you grasp the underlying principles and can confidently tackle similar problems. Acetic acid (CH3COOH) is a common weak acid, found in vinegar, and its behavior in water is governed by its Ka value. So, grab your calculator and let's dive into the world of acid-base chemistry!

Understanding the Problem

Before we start crunching numbers, let's make sure we understand what we're dealing with. We have a 0.273 M solution of acetic acid (CH3COOH), and we know its acid dissociation constant (Ka) is 1.8 x 10-5. The Ka value tells us how much the acid dissociates, or breaks apart, into ions in water. A small Ka value indicates a weak acid, meaning it doesn't dissociate much. Our goal is to find the pH of this solution, which is a measure of its acidity. Remember, pH is defined as -log[H+], where [H+] is the concentration of hydrogen ions in the solution. Therefore, to find the pH, we need to determine the hydrogen ion concentration ([H+]) produced by the acetic acid in water.

Setting Up the Equilibrium

Acetic acid (CH3COOH) reacts with water (H2O) in a reversible reaction, establishing an equilibrium. The balanced equation for this reaction is:

CH3COOH(aq) + H2O(l) ⇌ H3O+(aq) + CH3COO-(aq)

For simplicity, we often represent the hydronium ion (H3O+) as H+. So, the equation can be written as:

CH3COOH(aq) ⇌ H+(aq) + CH3COO-(aq)

To solve for the equilibrium concentrations, we use an ICE table (Initial, Change, Equilibrium). This table helps us organize the initial concentrations, the changes in concentration as the reaction reaches equilibrium, and the equilibrium concentrations themselves.

Constructing the ICE Table

Here's how we set up the ICE table for this problem:

CH3COOH H+ CH3COO-
Initial (I) 0.273 0 0
Change (C) -x +x +x
Equilibrium (E) 0.273 - x x x
  • Initial (I): We start with 0.273 M of acetic acid and no H+ or CH3COO- ions initially.
  • Change (C): As the acetic acid dissociates, its concentration decreases by 'x', and the concentrations of H+ and CH3COO- increase by 'x'.
  • Equilibrium (E): The equilibrium concentrations are the initial concentrations plus the changes.

Writing the Ka Expression

The acid dissociation constant (Ka) is the equilibrium constant for the dissociation of an acid. For acetic acid, the Ka expression is:

Ka = [H+][CH3COO-] / [CH3COOH]

Now, we can substitute the equilibrium concentrations from the ICE table into the Ka expression:

  1. 8 x 10-5 = (x)(x) / (0.273 - x)

Solving for x

We now have an equation with one unknown, 'x', which represents the equilibrium concentration of H+. To solve for 'x', we need to make an assumption to simplify the equation. Since acetic acid is a weak acid and its Ka is small, we can assume that 'x' is much smaller than the initial concentration of acetic acid (0.273 M). In other words, we assume that 0.273 - x ≈ 0.273. This assumption simplifies the equation to:

  1. 8 x 10-5 = x2 / 0.273

Now, we can solve for x:

x2 = (1.8 x 10-5) * 0.273 x2 = 4.914 x 10-6 x = √(4.914 x 10-6) x = 0.002217 M

Checking the Assumption

Before we proceed, it's crucial to check if our assumption (0.273 - x ≈ 0.273) was valid. To do this, we calculate the percent ionization:

Percent Ionization = (x / Initial Concentration of Acid) * 100% Percent Ionization = (0.002217 / 0.273) * 100% Percent Ionization = 0.812%

Since the percent ionization is less than 5%, our assumption is valid. If the percent ionization were greater than 5%, we would need to use the quadratic formula to solve for 'x' more accurately.

Calculating the pH

Now that we have the value of 'x', which represents the equilibrium concentration of H+, we can calculate the pH:

pH = -log[H+] pH = -log(0.002217) pH = 2.65

Therefore, the pH of a 0.273 M acetic acid solution is approximately 2.65.

Alternative Method: Using the Quadratic Formula (When the Assumption Fails)

If the percent ionization is greater than 5%, our simplifying assumption is invalid, and we must use the quadratic formula to solve for 'x'. Let's revisit our Ka expression:

  1. 8 x 10-5 = x2 / (0.273 - x)

Multiplying both sides by (0.273 - x), we get:

  1. 8 x 10-5(0.273 - x) = x2

Expanding and rearranging the equation into the standard quadratic form (ax2 + bx + c = 0):

x2 + (1.8 x 10-5)x - (1.8 x 10-5)(0.273) = 0 x2 + (1.8 x 10-5)x - 4.914 x 10-6 = 0

Now we can use the quadratic formula to solve for x:

x = (-b ± √(b2 - 4ac)) / 2a

Where:

a = 1 b = 1.8 x 10-5 c = -4.914 x 10-6

Plugging in the values:

x = (-1.8 x 10-5 ± √((1.8 x 10-5)2 - 4(1)(-4.914 x 10-6))) / 2(1)

x = (-1.8 x 10-5 ± √(3.24 x 10-10 + 1.9656 x 10-5)) / 2

x = (-1.8 x 10-5 ± √(1.9656324 x 10-5)) / 2

x = (-1.8 x 10-5 ± 0.004433) / 2

We will get two possible values for x:

x1 = (-1.8 x 10-5 + 0.004433) / 2 = 0.0022075 x2 = (-1.8 x 10-5 - 0.004433) / 2 = -0.0022255

Since concentration cannot be negative, we discard the negative value. Therefore, x = 0.0022075 M.

Now we calculate the pH:

pH = -log[H+] pH = -log(0.0022075) pH ≈ 2.655

As you can see, the pH value obtained using the quadratic formula is very close to the value we obtained using the simplifying assumption. This reinforces the validity of the assumption when the percent ionization is low. However, it's crucial to understand when the assumption is appropriate and when the quadratic formula is necessary for accurate results.

Key Takeaways

  • Calculating the pH of a weak acid solution involves setting up an equilibrium expression and using an ICE table.
  • The Ka value is crucial for determining the extent of dissociation of the acid.
  • A simplifying assumption (0. - x ≈ [Acid]initial) can be made if the percent ionization is less than 5%. Always check this assumption!
  • If the assumption is invalid, use the quadratic formula to solve for 'x' accurately.
  • The pH is calculated using the formula pH = -log[H+], where [H+] is the equilibrium concentration of hydrogen ions.

By understanding these principles and practicing with various examples, you'll become proficient in calculating the pH of weak acid solutions.

Practice Problems

To solidify your understanding, try solving these practice problems:

  1. Calculate the pH of a 0.15 M solution of formic acid (HCOOH), given that Ka = 1.8 x 10-4.
  2. Calculate the pH of a 0.50 M solution of hypochlorous acid (HOCl), given that Ka = 3.0 x 10-8.

Remember to set up the ICE table, write the Ka expression, solve for 'x', check your assumption, and then calculate the pH. Good luck!

For further learning and a deeper dive into acid-base chemistry, explore resources at Khan Academy.