Problem 65 - Chemistry: Principles and Reactions

Question

The activation energy for the reaction involved in the souring of raw milk is \(75 \mathrm{~kJ}\). Milk will sour in about eight hours at \(21^{\circ} \mathrm{C}\left(70^{\circ} \mathrm{F}=\right.\) room temperature). How long will raw milk last in a refrigerator maintained at \(5^{\circ} \mathrm{C}\) ? Assume the rate constant to be inversely related to souring time.

Step-by-Step Solution

Answer

Answer: Approximately 81.56 hours, or about 3.4 days.

Step 1: Convert temperatures to Kelvin

To work with absolute temperatures, we need to convert the given temperatures in Celsius to Kelvin. To do so, simply add 273.15 to the Celsius temperatures. $$ T_1 = 21^{\circ}\mathrm{C} + 273.15 = 294.15\mathrm{K} $$ $$ T_2 = 5^{\circ}\mathrm{C} + 273.15 = 278.15\mathrm{K} $$

Step 2: Use the Arrhenius equation to find the relationship between rate constants and temperature

We will use the following form of the Arrhenius equation: $$ \frac{k_2}{k_1} = \exp \left(\frac{-E_\mathrm{a}}{R} \left(\frac{1}{T_2} - \frac{1}{T_1}\right) \right) $$ Where \(k_1\) is the rate constant at temperature \(T_1\), \(k_2\) is the rate constant at temperature \(T_2\), \(E_\mathrm{a}\) is the activation energy, and \(R\) is the ideal gas constant (8.314 J/mol K). We are given that \(E_\mathrm{a} = 75\mathrm{kJ/mol}\) which is equal to 75,000 J/mol, and since the rate constant \(k\) is inversely related to souring time: $$ \frac{t_1}{t_2} = \exp \left(\frac{-75,000}{8.314} \left(\frac{1}{278.15} - \frac{1}{294.15}\right) \right) $$

Step 3: Calculate the souring time at the refrigerator temperature

We are given that \(t_1\), the souring time at \(21^{\circ}\mathrm{C}\), is 8 hours. We'll substitute this value and solve for \(t_2\), the souring time at \(5^{\circ}\mathrm{C}\): $$ \frac{8}{t_2} = \exp \left(\frac{-75,000}{8.314} \left(\frac{1}{278.15} - \frac{1}{294.15}\right) \right) $$ Solve for \(t_2\): $$ t_2 = \frac{8}{\exp \left(\frac{-75,000}{8.314} \left(\frac{1}{278.15} - \frac{1}{294.15}\right) \right)} $$ When we plug the numbers into the equation, we get: $$ t_2 \approx 81.56 \mathrm{~hours} $$ So, raw milk will last about 81.56 hours, or approximately 3.4 days, in a refrigerator maintained at \(5^{\circ}\mathrm{C}\).

Key Concepts

Arrhenius Equation

The Arrhenius Equation is a fundamental formula that chemists use to understand how reaction rates vary with temperature. It is expressed as: \[ k = A e^{-\frac{E_\text{a}}{RT}} \]where:

• **\(k\)** is the rate constant, indicating the speed of the reaction.
• **\(A\)** is the pre-exponential factor, related to the frequency of collisions and orientation of molecules.
• **\(E_\text{a}\)** is the activation energy, the energy barrier that must be overcome for a reaction to occur.
• **\(R\)** is the universal gas constant, approximately 8.314 J/mol K.
• **\(T\)** is the temperature in Kelvin.

This equation essentially tells us that higher temperatures provide more energy to molecules, increasing the likelihood that collisions offer enough energy to surpass the activation energy. In our exercise, we use a rearranged form of the equation to compare rate constants (and thus, reaction times) at two different temperatures.

Rate Constant

The rate constant, denoted as **\(k\)**, is crucial in determining the speed of a chemical reaction. In our context, it directly relates to the time it takes for milk to sour. A greater rate constant means the milk will sour faster.

It is important to remember that the rate constant depends heavily on temperature, which the Arrhenius Equation helps us understand. In our exercise, the problem assumes that the souring time of milk is inversely proportional to the rate constant, meaning that as the rate constant increases, the time taken for milk to sour decreases.
Therefore, understanding how rate constants change is essential for predicting how changes in conditions, such as temperature drops in a refrigerator, affect reaction times like milk souring.

Temperature Conversion

Temperature conversion is a necessary step when using the Arrhenius Equation because the equation requires temperature inputs in Kelvin, not Celsius or Fahrenheit. Kelvin is the absolute temperature scale, which makes it ideal for scientific calculations involving energy and kinetics.

Converting from Celsius to Kelvin is straightforward: simply add 273.15 to the Celsius temperature. For example:

• **21°C** becomes **294.15K**.
• **5°C** becomes **278.15K**.

This simple conversion allows us to work with temperatures in a way that fits the formulas used in reaction kinetics, ensuring calculations regarding reaction rates are accurate and meaningful.

Reaction Kinetics

Reaction kinetics involves studying the rates of chemical processes and how different conditions affect these rates. This branch of chemistry helps us understand how quickly reactions occur and what factors influence this speed.
Factors that affect reaction rates include:

• **Temperature**: Higher temperatures typically increase reaction rates as they provide more energy for particles to collide.
• **Concentration of reactants**: Greater concentrations often lead to higher reaction rates due to the increased likelihood of particle collisions.
• **Catalysts**: These substances alter reaction pathways and can significantly speed up or slow down reaction rates without being consumed.

In the exercise, reaction kinetics come into play as we consider how temperature changes (like refrigerating milk) affect the souring process, allowing us to make predictions about shelf life.