Question

Aspirin has the structural formula At body temperature $\left({37}^{\circ }\mathrm{C}\right),K_a$ for aspirin equals $3\times {10}^{-5}$. If two aspirin tablets, each having a mass of $325\mathrm{mg}$, are dissolved in a full stomach whose volume is $1\mathrm{L}$ and whose $\mathrm{pH}$ is 2, what percent of the aspirin is in the form of neutral molecules?

Short answer

Approximately 99.97% of the aspirin is in the form of neutral molecules when two tablets are dissolved in a full stomach with a volume of 1 L and pH of 2.

Step-by-step solution

Write the expression for K_a

K_a is the acid dissociation constant, which tells us the degree to which an acid can dissociate into its ions. For aspirin, the dissociation can be represented as: $$HA\rightleftharpoons H^{+}+A^{-}$$ Given that the K_a for aspirin at 37°C is $3\times {10}^{-5}$, we can write the expression for K_a as: $$K_a=\frac{[H^{+}][A^{-}]}{[HA]}$$

Calculate the concentration of aspirin

Two aspirin tablets, each having a mass of 325 mg, are dissolved in a stomach with a volume of 1 L. Therefore, the total mass of aspirin is: $$2\times 325\text{mg}$$ Convert the mass to grams: $2\times 325\text{mg}=650\text{mg}=0.65\text{g}$ Now, find the molarity (moles per liter) of the aspirin. The molar mass of aspirin is $\approx 180\text{g/mol}$, so divide the mass by the molar mass and the volume to get the concentration: $$C_{HA}=\frac{0.65\text{g}}{180\text{g/mol}\times 1\text{L}}=0.00361\text{M}$$

Calculate the concentration of H+ ions

We are given the pH of the solution, which is 2. The pH is related to the concentration of H+ ions by the following relationship: $$pH=-{\log }_{10}([H^{+}])$$ Solving for $[H^{+}]$, we get: $$[H^{+}]={10}^{-pH}={10}^{-2}=0.01\text{M}$$

Determine the equilibrium concentrations

Let x represent the change in concentration of aspirin when it dissociates: $$[HA]=0.00361-x\text{M}$$ $$[A^{-}]=x\text{M}$$ $$[H^{+}]=0.01+x\text{M}$$ Now, substitute these expressions into the K_a expression and solve for x: $$3\times {10}^{-5}=\frac{(0.01+x)(x)}{(0.00361-x)}$$

Solve for x

Since $K_a$ is small, we can assume that the change in concentration (x) is also small and therefore negligible compared to the initial concentrations of aspirin and H+ ions. Therefore, we can simplify the equation as: $$3\times {10}^{-5}\approx \frac{(0.01)(x)}{0.00361}$$ Now, solve for x: $$x\approx 1.08\times {10}^{-6}\text{M}$$

Calculate the percentage of neutral aspirin molecules

We found the change in concentration, x, which represents the concentration of A- ions formed due to the dissociation of aspirin. The percentage of neutral aspirin molecules, or HA, in the solution can be calculated using the following formula: $$\mathrm{\%}\text{of}HA=\frac{[HA]}{[HA]+[A^{-}]}\times 100$$ Substitute the equilibrium concentrations from Step 4: $$\mathrm{\%}\text{of}HA=\frac{0.00361-x}{0.00361}\times 100=\frac{0.00361-1.08\times {10}^{-6}}{0.00361}\times 100$$ Finally, calculate the percentage of neutral aspirin molecules: $$\mathrm{\%}\text{of}HA\approx 99.97\mathrm{\%}$$ Thus, approximately 99.97% of the aspirin is in the form of neutral molecules.

Key concepts

Aspirin Chemistry

Understanding aspirin chemistry is crucial when examining its behavior in different environments, such as the human stomach. Aspirin, also known as acetylsalicylic acid (ASA), is a salicylate drug often used to alleviate pain, reduce fever, and minimize inflammation. Chemically, it can be represented by the molecular formula C₉H₈O₄.

When aspirin enters the acidic environment of the stomach, it can exist either as neutral molecules (HA) or as ions following its dissociation into H⁺ (protons) and A^- (the conjugate base of aspirin). The extent of this dissociation is governed by the acid dissociation constant, K_a, which at body temperature for aspirin is given as 3 × 10^-5. This relatively small value indicates that aspirin is a weak acid and will not dissociate significantly in the acidic environment of the stomach.

Chemical Equilibrium

Chemical equilibrium plays a fundamental role in the process of acid dissociation. It is the point in a reversible chemical reaction at which the rate of the forward reaction equals the rate of the reverse reaction, resulting in no overall change in the concentrations of reactants and products over time.

For aspirin, the equilibrium can be represented by the equation:
HA ⇌ H⁺ + A^-.
The reaction quotient (Q) may become equal to the acid dissociation constant (K_a) at equilibrium, and at this point, the concentrations of the species involved do not change further. It is crucial to understand that even when the reaction has reached equilibrium, the reaction is still proceeding; however, the rates of forward and reverse reactions are now the same. This K_a value helps to predict the degree of dissociation of aspirin under various conditions, aiding in calculating the percentage of aspirin that remains undissociated in solution.

pH Calculation

pH calculation is critical for understanding the acidity or basicity of a solution. It is succinctly defined as the negative logarithm of the hydrogen ion concentration in a solution (pH = -log₁₀[H⁺]). The pH scale is usually measured from 0 to 14, with 7 being neutral, values below 7 acidic, and values above 7 basic.

In the context of the aspirin problem, knowing the stomach's pH, which is 2, allows us to deduce the concentration of H⁺ ions to be 0.01M. This high concentration of protons in the stomach will dominate the equilibrium shifting the equilibrium to the left, favoring the undissociated form of the aspirin. Because aspirin is a weak acid and the stomach is highly acidic (with a low pH), we can anticipate that the majority of the aspirin will indeed remain in its neutral form. The subsequent calculations of equilibrium concentrations show that only a tiny fraction of aspirin dissociates, leaving the vast majority in its undissociated, neutral molecule form.