Problem 1
Question
A student studies the rate at which aspirin decomposes by the reaction: $$ \mathrm{Asp}+\mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{Sal}+\text { Acetic Acid } $$ She weighs out \(58.5 \mathrm{mg}\) of aspirin and dissolves it, making \(10 \mathrm{mL}\) of solution in water. She heats the solution for 5 minutes at \(90^{\circ} \mathrm{C}\), and finds that about \(10 \%\) of the aspirin is converted to salicylic acid and acetic acid. a. How many moles of aspirin are in the initial solution? What is the molarity of the aspirin? MM aspirin \(=180 \mathrm{g}\) ____moles____ \(\mathbf{M}\) b. What is the rate of reaction? (See Eq. 1 on Data page.)____ moles/L min In a similar experiment with a smaller sample, she finds that once again about \(10 \%\) of the aspirin 0 decomposes in 5 minutes at \(100^{\circ} \mathrm{C}\). What is the order of the reaction? Why? d. What is the rate constant for the reaction? e. In another experiment she finds that the energy of activation \(E_{a}\) for the reaction is about \(90 \mathrm{kJ}\). How long would it take for \(10 \%\) of an aspirin sample to decompose in a person's stomach at body temcrature, \(37^{\circ} \mathrm{C}\) ____minutes
Step-by-Step Solution
Verified Answer
The moles of aspirin in the initial solution are approximately 3.25 × 10⁻⁴ mol, and the molarity is 0.0325 M. The rate of reaction is 6.5 × 10⁻⁵ mol/L min, and the order of reaction is zero-order. The rate constant for the reaction is 6.5 × 10⁻⁵ mol/L min. For 10% decomposition of an aspirin sample at body temperature (37°C), it would take approximately 22.26 minutes.
1Step 1: Calculate Moles of Aspirin in the Initial Solution
To find the moles of aspirin in the initial solution, we will use the formula:
Moles of Aspirin = (Mass of Aspirin) / (Molar Mass of Aspirin)
We are given the mass of aspirin as 58.5 mg and the molar mass as 180 g. First, we need to convert the mass to grams:
Mass of Aspirin = (58.5 mg) * (1g / 1000 mg) = 0.0585 g
Moles of Aspirin = (0.0585 g) / (180 g/mol) = 3.25 × 10⁻⁴ mol
2Step 2: Calculate the Molarity of the Aspirin Solution
To find the molarity, we will use the formula:
Molarity = (Moles of Aspirin) / (Volume of Solution in liters)
Volume of Solution = 10 mL = 0.01 L
Molarity = (3.25 × 10⁻⁴ mol) / (0.01 L) = 0.0325 M
3Step 3: Calculate the Rate of Reaction
To find the rate of reaction, we will use the formula:
Rate of Reaction = (moles of Aspirin converted) / (volume of solution * time)
10% of aspirin is converted, so:
Moles of Aspirin converted = 0.1 * (3.25 × 10⁻⁴ mol) = 3.25 × 10⁻⁵
Time = 5 minutes
Rate of Reaction = (3.25 × 10⁻⁵ mol) / (0.01 L * 5 min) = 6.5 × 10⁻⁵ mol/L min
4Step 4: Find the Order of Reaction
Since the reaction rate is independent of the initial aspirin concentration (10% decomposed in both experiments), it indicates that this reaction is zero-order.
5Step 5: Calculate the Rate Constant
For a zero-order reaction, the rate constant (k) can be found using the equation:
Rate of Reaction = k * (concentration)^0
Rate of Reaction = k
k = 6.5 × 10⁻⁵ mol/L min
6Step 6: Calculate the Time for 10% Decomposition at 37°C
Here, we will use the Arrhenius equation and then find the rate constant at 37°C:
k = A * exp(-Ea / (R * T))
where A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is the temperature in Kelvin.
Assuming A is constant, we can compare the rate constants of different temperatures:
k(37°C) / k(90°C) = exp(-(Ea / (R * 37°C + 273)) / (Ea / (R * 90°C + 273)))
k(37°C) = k(90°C) * exp(-(Ea / (R * 37°C + 273)) / (Ea / (R * 90°C + 273)))
k(37°C) = 6.5 × 10⁻⁵ mol/L min * exp(-(90 kJ) / (8.314 J/mol·K * (37°C + 273)) / (90 kJ) / (8.314 J/mol·K * (90°C + 273)))
k(37°C) = approx. 1.46 × 10⁻⁵ mol/L min
Now we can determine the time it takes for 10% decomposition at 37°C:
Rate of Reaction at 37°C = k(37°C)
1.46 × 10⁻⁵ mol/L min = (3.25 × 10⁻⁵ mol) / (0.01 L * time)
Time = (3.25 × 10⁻⁵ mol) / (0.01 L * 1.46 × 10⁻⁵ mol/L min)
Time ≈ 22.26 minutes
Key Concepts
Molarity CalculationZero-order ReactionArrhenius Equation
Molarity Calculation
Molarity is an essential concept in chemistry that helps us determine the concentration of a solute in a solution. It's defined as the number of moles of solute divided by the volume of the solution in liters. To calculate the molarity of a solution, you need to know two things:
- The number of moles of the solute.
- The volume of the solution in liters.
Zero-order Reaction
Zero-order reactions are unique because their rates are independent of the concentration of the reactants. This means that even if you change the concentration of the reactant, the rate of reaction remains constant. This behavior is observed in certain reactions where a limiting factor, such as enzyme availability or surface area, controls the reaction rate instead of reactant concentration.To identify a zero-order reaction, look for an experiment where the concentration decreases linearly over time, regardless of initial concentration. In our aspirin decomposition example, both trials show a consistent 10% conversion rate, which implies a zero-order reaction. The reaction rate formula for zero-order reactions is:\[\text{Rate of Reaction} = k\]where \(k\) is the rate constant. This means the rate constant is crucial for determining the reaction speed because the concentration is raised to the power of zero, thus not affecting the rate. Understanding zero-order kinetics is essential in fields like pharmacology and material science, where maintaining constant reaction rates is beneficial.
Arrhenius Equation
The Arrhenius equation is an important relationship in chemistry that provides insight into how temperature and activation energy affect reaction rates. It is expressed as:\[k = A \cdot e^{-\frac{E_a}{R \cdot T}}\]where:
- \(k\) is the rate constant.
- \(A\) is the pre-exponential factor (frequency of collisions factor).
- \(E_a\) is the activation energy in joules/mole.
- \(R\) is the gas constant (8.314 J/mol·K).
- \(T\) is the temperature in Kelvin.