Question

Consider the coagulation of a protein at \(100^{\circ} \mathrm{C}\). The first- order reaction has an activation energy of \(69 \mathrm{~kJ} / \mathrm{mol}\). If the protein takes 5.4 minutes to coagulate in boiling water at \(100^{\circ} \mathrm{C},\) then how long will it take to coagulate the protein at an altitude where water boils at \(87^{\circ} \mathrm{C}\) ?

Short answer

Given the coagulation time of a protein at 100°C and the activation energy of the reaction, we can use the Arrhenius equation to find the time it takes for the protein to coagulate at a lower temperature of 87°C. After converting temperatures to Kelvin and the given time to seconds, and plugging in the necessary values into the Arrhenius equation, we find the rate constant at 87°C to be approximately \(2.0333 \times 10^{-4} \text{s}^{-1}\). Using the first-order rate equation, we find the coagulation time at 87°C to be approximately 4915 seconds, or about 81.9 minutes.

Step-by-step solution

Convert temperatures to Kelvin

Before we work with the temperature values, we need to convert them from Celsius to Kelvin. To do this, add 273.15 to each of the given temperatures: \(100^{\circ} \mathrm{C} = 100 + 273.15 = 373.15 \mathrm{K}\) \(87^{\circ} \mathrm{C} = 87 + 273.15 = 360.15 \mathrm{K}\)

Convert time to seconds

The given coagulation time is in minutes, but we'll need it in seconds for our calculations. To convert minutes to seconds, multiply by 60: \(5.4 \, \text{min} = 5.4 \times 60 = 324 \, \text{s}\)

Use the Arrhenius equation

According to the Arrhenius equation, the rate constants of a reaction at two different temperatures can be related as follows: \[ k_2 = k_1 e^{ \frac{-E_a}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)} \] where \(k_1\) and \(k_2\) are the rate constants at temperatures \(T_1\) and \(T_2\), respectively, \(E_a\) is the activation energy, and \(R\) is the gas constant (\(8.314 \, \text{J} \text{mol}^{-1} \text{K}^{-1}\)). Since the reaction is first-order, the rate equation can be written as follows: \[ k = \frac{1}{t} \] where \(t\) is the coagulation time. Hence, we can compute the rate constants as \(k_1 = \frac{1}{324\, \text{s}}\) at 100°C and \(k_2\) as the unknown rate constant at 87°C.

Insert given values and solve for \(k_2\)

Now, we will plug the given values and calculated \(k_1\) into the Arrhenius equation and solve for \(k_2\): \[ k_2 = \frac{1}{324\, \text{s}} e^{ \frac{-69000 \, \text{J}\text{mol}^{-1}}{8.314\, \text{J}\text{mol}^{-1}\text{K}^{-1}} \left( \frac{1}{360.15\, \text{K}} - \frac{1}{373.15\, \text{K}} \right)} \] After solving the above expression, we get: \[ k_2 \approx 2.0333 \times 10^{-4} \text{s}^{-1} \]

Find the time it takes to coagulate at 87°C

We now have the rate constant, \(k_2\), for the reaction at 87°C. To find the coagulation time at this temperature, we use the first-order rate equation we derived earlier: \[ t_2 = \frac{1}{k_2} \] \[ t_2 = \frac{1}{2.0333 \times 10^{-4} \text{s}^{-1}} \] This gives us a coagulation time of approximately 4915 seconds, or about 81.9 minutes, at 87°C.

Key concepts

Arrhenius Equation

The Arrhenius equation is critical when it comes to understanding how temperature affects reaction rates. It describes the relationship between the rate constant, k, and temperature, T, using the activation energy, E_a, and a universal gas constant, R. In simplified terms, it explains that a higher temperature increases the reaction rate because it helps to overcome the activation energy barrier more effectively.

The equation is usually written as:
\[ k = A e^{\frac{-E_a}{RT}} \]
In this expression, A stands for the pre-exponential factor, a constant for each chemical reaction representing the collision frequency and orientation of reacting molecules. As the temperature rises, it's more likely that molecules will collide with enough energy to react, thus increasing the rate constant k. This equation is indispensable in predicting how a reaction's speed will change under different thermal conditions.

Activation Energy

Activation energy, often represented as E_a, is the minimum amount of energy that reacting particles must possess for a reaction to occur. Think of it as a hill that reactants must climb to transform into products. If the energy of the particles is below this threshold, the reaction will not proceed, irrespective of the concentration or presence of reactants.

The importance of activation energy lies in its influence on reaction rates. A lower activation energy means that more particles will have enough energy to react at a given temperature, speeding up the reaction. Conversely, a high activation energy results in a slower reaction as fewer particles have the sufficient energy to overcome this barrier.

First-Order Reaction

A first-order reaction is a type of chemical reaction where the rate is directly proportional to the concentration of one reactant. This implies that if you double the concentration of the reactant, the reaction rate also doubles. In the realm of kinetics, such reactions have a simple linear relationship which is mathematically expressed as:
\[ \text{Rate} = k [\text{A}] \]
Here, [A] represents the concentration of the reactant and k is the rate constant, which is unique for every reaction at a given temperature. The hallmark of first-order reactions is that their rate depends on the concentration of only one reactant, making them simpler to analyze compared to reactions of higher order.

Rate Constants

Rate constants play a pivotal role in chemical kinetics. Represented by the letter k, they are a measure of the speed at which a reaction proceeds. Specifically, for a first-order reaction, the rate constant k is the proportionality factor that connects the reaction rate to the reactant concentration.

It's crucial to understand that the value of the rate constant is determined by factors such as temperature, presence of a catalyst, and the intrinsic properties of the reactants. The units of rate constants also vary with the order of the reaction; for a first-order reaction, the units are typically s^-1 or min^-1. A larger rate constant indicates a faster reaction, suggesting it takes less time for the reactants to turn into products.