A typical aspirin tablet contains 323 mg acetylsalicylic acid [tex]$\left( HC _9 H _7 O _4\right)$[/tex]. Calculate the pH of a solution that is prepared by dissolving two aspirin tablets in enough water to make one cup (237 mL) of solution. Assume the aspirin tablets are pure acetylsalicylic acid, [tex]$K_{ a }=3.3 \times 10^{-4}$[/tex].
Answer (2)
Calculate the mass of acetylsalicylic acid: ma ss = 2 \t × 323 mg = 646 mg = 0.646 g .
Calculate the moles of acetylsalicylic acid: m o l es = \t 180.17 g/mol 0.646 g = 0.003586 mol .
Calculate the molar concentration: M = \t 0.237 L 0.003586 mol = 0.01513 M .
Calculate the pH: p H = − lo g 10 ( 0.002076 ) = 2.68 . The final answer is 2.68 .
Explanation
Problem Setup and Given Information We are given that a typical aspirin tablet contains 323 mg of acetylsalicylic acid ( H C 9 H 7 O 4 ). We need to calculate the pH of a solution made by dissolving two aspirin tablets in 237 mL of water. The acid dissociation constant, K a , for acetylsalicylic acid is 3.3 × 1 0 − 4 . We assume the tablets are pure acetylsalicylic acid.
Calculating Moles of Acetylsalicylic Acid First, we calculate the total mass of acetylsalicylic acid in the solution. Since we have two tablets, the total mass is: ma ss = 2 × 323 mg = 646 mg Next, we convert this mass to grams: ma s s g r am s = 1000 mg/g 646 mg = 0.646 g Now, we calculate the number of moles of acetylsalicylic acid. The molecular weight of acetylsalicylic acid ( H C 9 H 7 O 4 ) is: M W = 9 ( 12.01 ) + 8 ( 1.01 ) + 4 ( 16.00 ) = 108.09 + 8.08 + 64.00 = 180.17 g/mol So, the number of moles is: m o l es = 180.17 g/mol 0.646 g = 0.003586 mol
Calculating Molar Concentration Next, we calculate the molar concentration of the solution. The volume of the solution is 237 mL, which is 0.237 L. Therefore, the molar concentration is: M = 0.237 L 0.003586 mol = 0.01513 M Now, we set up an ICE table for the dissociation of acetylsalicylic acid: H C 9 H 7 O 4 ( a q ) + H 2 O ( l ) ⇌ H 3 O + ( a q ) + C 9 H 7 O 4 − ( a q ) Initial: 0.01513 M 0 0 Change: -x +x +x Equilibrium: 0.01513 - x x x
Setting up the ICE Table and Ka Expression We write the expression for K a :
K a = [ H C 9 H 7 O 4 ] [ H 3 O + ] [ C 9 H 7 O 4 − ] = 0.01513 − x x 2 = 3.3 × 1 0 − 4 We can solve for x using the quadratic formula or by assuming that x is small compared to the initial concentration of the acid. Let's assume x is small, so 0.01513 − x ≈ 0.01513 :
0.01513 x 2 = 3.3 × 1 0 − 4 x 2 = ( 3.3 × 1 0 − 4 ) ( 0.01513 ) = 4.9929 × 1 0 − 6 x = 4.9929 × 1 0 − 6 = 0.002235 M Since x is about 15% of the initial concentration, the approximation is not very good, so we should use the quadratic formula. However, to avoid errors, we will proceed with the approximation.
Calculating pH Now, we calculate the pH: p H = − lo g 10 ([ H 3 O + ]) = − lo g 10 ( 0.002235 ) = 2.65 However, let's solve this more accurately using the quadratic formula: x 2 + 3.3 × 1 0 − 4 x − ( 3.3 × 1 0 − 4 ) ( 0.01513 ) = 0 x 2 + 0.00033 x − 4.9929 × 1 0 − 6 = 0 Using the quadratic formula: x = 2 a − b ± b 2 − 4 a c = 2 ( 1 ) − 0.00033 ± ( 0.00033 ) 2 − 4 ( 1 ) ( − 4.9929 × 1 0 − 6 ) x = 2 − 0.00033 ± 1.089 × 1 0 − 7 + 1.99716 × 1 0 − 5 = 2 − 0.00033 ± 2.00805 × 1 0 − 5 x = 2 − 0.00033 ± 0.004481 Since x must be positive: x = 2 − 0.00033 + 0.004481 = 2 0.004151 = 0.002076 M p H = − lo g 10 ( 0.002076 ) = 2.68
Final Answer The pH of the solution is approximately 2.68.
Examples
Aspirin is a common medication used to relieve pain and reduce fever. Understanding the pH of aspirin solutions is important in pharmaceutical formulations and drug delivery. For example, the solubility and absorption of aspirin can be affected by the pH of the environment. By calculating the pH of an aspirin solution, we can better understand its behavior in the body and optimize its therapeutic effects. This knowledge is also useful in quality control during the manufacturing process to ensure that the final product meets the required specifications.
The pH of a solution prepared by dissolving two aspirin tablets (646 mg) in 237 mL of water is approximately 2.65. This calculation involves determining the molar concentration and applying the acid dissociation constant. The solution is considered acidic because acetylsalicylic acid is a weak acid.
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