Question

(Postacidification in yogurt) Yogurt is produced by adding two strains of bacteria (Lactobacillus bulgaricus and Streptococcus thermophilus) to pasteurized milk. At temperatures of \(110^{\circ} \mathrm{F}\), the bacteria grow and produce lactic acid. The acid contributes flavor and causes the proteins to coagulate, giving the characteristic properties of yogurt. When sufficient acid has been produced (about \(0.90 \%\) ), the yogurt is cooled and stored until eaten by consumers. A lactic acid level of \(1.10 \%\) is the limit of acceptability. One limit on the shelf life of yogurt is "postacidification," or continued production of acid by the yogurt cultures during storage. The table that follows shows acid production (\% lactic acid) in yogurt versus time at four different temperatures. Acid production by yogurt cultures is a complex biochemical process. For the purpose of this problem, assume that acid production follows first-order kinetics with respect to the consumption of lactose in the yogurt to produce lactic acid. At the start of acid production the lactose concentration is about \(1.5 \%,\) the bacteria concentration is \(10^{11}\) cells/dm \(^{3}\), and the acid concentration at which all metabolic activity ceases is \(1.4 \%\) lactic acid. (a) Determine the activation energy for the reaction. (b) How long would it take to reach \(1.10 \%\) acid at \(38^{\circ} \mathrm{F}\) ? (c) If you left yogurt out at room temperature, \(77^{\circ} \mathrm{F}\), how long would it take to reach 1.10\% lactic acid? (d) Assuming that the lactic acid is produced in the stationary state, do the data fit any of the modules developed in this chapter? [Problem developed by General Mills, Minneapolis, Minnesota] (e) List ways you can work this problem incorrectly. (f) How could you make this problem more difficult?

Short answer

The activation energy (Ea) for the acid production reaction can be determined using the Arrhenius equation and first-order kinetics. By calculating the rate constant at different temperatures, the time required to reach 1.10% lactic acid concentration at \(38^{\circ} \mathrm{F}\) and \(77^{\circ} \mathrm{F}\) can be estimated, which involves solving the first-order reaction rate expression for time (t). Incorrect approaches to this problem might include misapplying the first-order reaction kinetics or Arrhenius equation or using incorrect initial or final concentrations for calculations. The problem could be made more difficult by considering the impact of pH changes on the reaction rate or involving more complex kinetics like second-order or reversible reactions.

Step-by-step solution

Determine the Activation Energy

To determine the activation energy (Ea) for the reaction, the Arrhenius equation will be used: \[ k=Ae^{-Ea/RT} \] where \(k\) is the rate constant, \(A\) is the pre-exponential factor (representing the frequency of collisions between reactant molecules), \(Ea\) is the activation energy, \(R\) is the universal gas constant, and \(T\) is the temperature in Kelvin. Since rate constant calculations are not provided in the problem, we need to first calculate the rate constant at the mentioned temperature (\(110^{\circ} \mathrm{F} = 43^{\circ} \mathrm{C} = 316 K\)) utilizing the first order kinetics model, and then rearrange the Arrhenius equation to calculate \(Ea\).

Calculating Time to Reach 1.10% Acid at 38°F

To estimate the time it would take to reach \(1.10\% \) acid at \(38^{\circ} \mathrm{F}\), we need to first convert the temperature to Kelvin (i.e., \(38^{\circ} \mathrm{F} = 3.33^{\circ} \mathrm{C} = 276.48 K\)). Then we will apply the first-order kinetics reaction rate expression \(c = c_{0}e^{kt}\) (where \(c_{0}\) and \(c\) are the initial and final concentrations respectively, \(k\) is the rate constant and \(t\) is time) and rearrange it to solve for \(t\).

Calculating Time to Reach 1.10% Lactic Acid at 77°F

To calculate the time it takes to reach \(1.10\%\) lactic acid at \(77^{\circ} \mathrm{F}\), we again convert the temperature to Kelvin (i.e., \(77^{\circ} \mathrm{F} = 25^{\circ} \mathrm{C} = 298.15 K\)). Then we apply the first-order kinetics reaction rate equation and manipulate for \(t\).

Analyzing the Fit of the Data

To comment on whether the data fits any models developed, additional information on the data and models would have been necessary. If models of first-order reactions were developed, this specific scenario could potentially fall into such a model.

Listing Possible Incorrect Approaches

Several ways to work this problem incorrectly might include: using incorrect units, failing to convert temperatures to Kelvin, misapplying the first-order reaction kinetics or Arrhenius equation, using incorrect initial or final concentrations for calculations.

Ways to Make the Problem More Difficult

This problem could be further complicated by adding multiple reactants contributing to the acid production, considering the impact of pH changes on the reaction rate, or involving more complex kinetics like second-order or reversible reactions.

Key concepts

Arrhenius Equation

The Arrhenius equation is a formula that connects the rate of chemical reactions with temperature. It's given by the expression \[ k = Ae^{-Ea/RT} \.\]
In this equation, \(k\) represents the rate constant, reflecting the speed at which a reaction proceeds. The rate constant increases with temperature, indicating reactions occur faster at higher temperatures. \(A\) is the pre-exponential factor, which is related to the frequency of collisions between reactant molecules—in essence, it's a measure of how often particles collide with the right orientation to react. \(Ea\) stands for activation energy, the minimum energy required for a reaction to occur when molecules collide. \(R\) is the universal gas constant, and \(T\) is the temperature in Kelvin.

When using the Arrhenius equation to calculate the activation energy for lactic acid production in yogurt, one would plot the natural log of the rate constant (ln(k)) against the reciprocal of the temperature (1/T) and determine \(Ea\) from the slope of this plot. The steeper the slope, the higher the activation energy, implying greater temperature sensitivity.

First-Order Kinetics

First-order kinetics describes a scenario where the rate at which a reaction proceeds is directly proportional to the concentration of one of the reactants. This relationship is mathematically represented as \[ c = c_0e^{-kt} \.\]
Here, \(c_0\) is the initial concentration of the reactant, \(c\) is the concentration at time \(t\), and \(k\) is the first-order rate constant.

Applying first-order kinetics to the lactic acid production in yogurt, the consumption of lactose to produce lactic acid can be modeled to predict how the concentration of the acid changes over time. This modeling is critical for estimating the shelf life of yogurt, as excessive lactic acid results in over-acidification, rendering the yogurt unacceptable. Calculating the time to reach a certain concentration, such as 1.10%, relies on knowing or estimating the value of \(k\) and involves manipulating the first-order reaction equation accordingly.

Activation Energy

Activation energy (\(Ea\)) is a pivotal concept in chemical kinetics and represents the minimum amount of energy that reacting molecules must have for a reaction to occur upon collision. It's an energy barrier that reactants must overcome to transform into products.

The amount of activation energy required affects how fast a reaction takes place: the higher the activation energy, the slower the reaction rate, as fewer molecules have enough energy to react when they collide. In the context of lactic acid production in yogurt, the activation energy would determine how swiftly the bacteria can convert lactose into lactic acid at different temperatures. Generally, producers aim to control the activation energy to maximize the rate of acid production for optimal yogurt texture and flavor within the desired timescale.

Lactic Acid Production

Lactic acid production in yogurt is an essential process carried out by specific bacterial cultures that ferment lactose, the sugar found in milk. This biochemical process not only contributes to the characteristic tangy flavor of yogurt but also leads to the coagulation of milk proteins, resulting in yogurt's thick texture.

During storage, bacteria continue to produce lactic acid in a phenomenon known as postacidification. Monitoring and controlling this continued acidification are crucial to ensure the product's taste and texture remain within acceptable limits before consumption. By understanding and applying concepts such as the Arrhenius equation and first-order kinetics, dairy technologists can predict and manage the rate of lactic acid production, thus prolonging the yogurt's shelf life and maintaining its quality.