Problem 79 - Chemistry and Chemical Reactivity

Question

Egg protein albumin is precipitated when an egg is cooked in boiling \(\left(100^{\circ} \mathrm{C}\right)\) water. \(E_{\mathrm{a}}\) for this first- order reaction is \(52.0 \mathrm{kJ} / \mathrm{mol} .\) Estimate the time to prepare a 3 -minute egg at an altitude at which water boils at \(90^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

The estimated time to prepare a 3-minute egg at 90°C is approximately 7.8 minutes.

Step 1: Understand the Reaction Mechanism

The exercise is about estimating the cooking time for an egg where the protein albumin undergoes a first-order reaction with a given activation energy \(E_a = 52.0 \text{kJ/mol}\). The boiling point of water changes due to altitude, affecting the reaction rate.

Step 2: Arrhenius Equation

Apply the Arrhenius equation which statistically explains how the rate constant \(k\) of a reaction depends on temperature \(T\) and activation energy \(E_a\). This equation is: \[ k = Ae^{-\frac{E_a}{RT}} \] where \(A\) is the pre-exponential factor, and \(R = 8.314 \text{ J/mol·K}\) is the universal gas constant.

Step 3: Express the Rate Constants

The rate constant \(k\) is different at different temperatures. Let \(k_1\) be the rate constant at \(100^{\circ} \text{C} (373 \text{ K})\) and \(k_2\) at \(90^{\circ} \text{C} (363 \text{ K})\). The ratio \(\frac{k_2}{k_1}\) can be derived from the Arrhenius equation: \[ \ln\left(\frac{k_2}{k_1}\right) = -\frac{E_a}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right) \]

Step 4: Calculate the Ratio of Rate Constants

Substitute values into the derived formula: \(T_1 = 373 \text{ K} \), \(T_2 = 363 \text{ K} \), and \(E_a = 52000 \text{ J/mol}\). Substitute and calculate:\[ \ln\left(\frac{k_2}{k_1}\right) = -\frac{52000}{8.314}\left(\frac{1}{363} - \frac{1}{373}\right)\]Solve for \(\ln\left(\frac{k_2}{k_1}\right)\).

Step 5: Estimated Reaction Time

As the reaction rate \(k\) is reciprocally related to the time for a first-order reaction, the time ratio is inverted:\[ \frac{t_2}{t_1} = \frac{k_1}{k_2} \]If \(t_1 = 3\) minutes at \(100^{\circ} \text{C}\), calculate \(t_2\) using the equation from Step 4.

Step 6: Compute and Interpret Results

Calculate the reaction time \(t_2\) at \(90^{\circ} \text{C}\) using the above steps. This will show the cooking time needed at the reduced boiling point of water. Use the calculated ratio from Step 4 to adjust the 3-minute time accordingly.

Key Concepts

Reaction Mechanism

In chemistry, the term "reaction mechanism" refers to a detailed step-by-step description of how a chemical reaction occurs. It includes the order of events, the rearrangement of bonds, and the intermediate structures formed during the reaction. For a reaction to occur, reactants must collide with sufficient energy to surpass an energy barrier known as the activation energy. This process forms the reactants into products.

In the context of the exercise, the cooking of an egg is depicted as a chemical reaction. Specifically, egg protein albumin changes state when subjected to heat. Here, the mechanism highlights how this transformation follows a first-order kinetic pattern. The reaction involves the breaking of specific bonds in the egg's proteins, which results in them being "cooked" or denatured due to heat impacting the boiled water.

• The process is temperature-dependent, meaning hotter conditions enhance the reaction speed.
• The height at which you boil the egg can affect the temperature, altering the reaction mechanism.

Activation Energy

Activation energy is the minimum amount of energy required for a chemical reaction to occur. It acts as a barrier that the reactants need to overcome to transform into products. This concept is pivotal in understanding why reactions happen faster at higher temperatures. More heat provides more energy to surpass this activation threshold.

In the given exercise, the activation energy (\(E_a\)) is specified as \(52.0 \text{kJ/mol}\). This means each mole of the reacting albumin molecules requires this amount of energy to transition to the cooked state. At higher temperatures, like 100°C, molecules have more kinetic energy to overcome the activation energy, thus reacting more quickly compared to the lower temperature of 90°C.

• An increase in temperature generally decreases the time needed for the reaction as it effectively reduces the relative impact of activation energy.
• Understanding activation energy helps in determining the feasibility and rate of reaction under varying conditions.

Rate Constant

The rate constant (\(k\)) is a key factor in the study of chemical kinetics. It quantifies the speed of a reaction. The rate constant encapsulates how factors such as temperature and catalyst presence can influence the rate of a reaction.

For a first-order reaction involving albumin, \(k\) becomes temperature-dependent. By applying the Arrhenius equation, one can estimate how \(k\) changes with temperature, which further impacts the time required for a reaction to reach completion. The rate constant is crucial for predicting how quickly the egg will cook at the given temperature.

• The Arrhenius equation, \(k = Ae^{-\frac{E_a}{RT}}\), relates \(k\) to temperature (\(T\)) and activation energy (\(E_a\)).
• The expression for the rate constant helps determine reaction time and its adjustment according to temperature shifts.

First-Order Reaction

A first-order reaction is one where the rate of the reaction depends linearly on the concentration of one reactant. As the concentration of the reactant decreases over time, the reaction proceeds to completion more slowly. In mathematical terms, the rate can be expressed as:\[ Rate = k[A] \]where \(k\) is the rate constant and \([A]\) is the concentration of the reactant.

In this exercise, the denaturation of the protein albumin in the egg follows a first-order reaction model. This implies that cooking time depends not only on the activation energy and temperature but also on the initial state of the egg proteins. Changes in environmental conditions such as temperature will change the reaction's completion time by affecting \(k\).

• The first-order kinetic model provides a straightforward approach to estimate how duration varies with temperature change, affecting egg cooking times based on boiling points adjusted for altitude.
• Knowing it's a first-order reaction allows for using logarithmic calculations to derive cooking times under different thermal conditions.