Question

When arterial blood enters a tissue capillary, it exchanges oxygen and carbon dioxide with its environment, as shown in this diagram. The kinetics of this deoxygenation of hemoglobin in blood was studied with the aid of a tubular reactor by Nakamura and Staub \([J . \text { Physiol.}, 173,161]\). Although this is a reversible reaction, measurements were made in the initial phases of the decomposition so that the reverse reaction could be neglected. Consider a system similar to the one used by Nakamura and Staub: the solution enters a tubular reactor \((0.158 \mathrm{cm}\) in diameter) that has oxygen electrodes placed at \(5 \cdot \mathrm{cm}\) intervals down the tube. The solution flow rate into the reactor is \(19.6 \mathrm{cm}^{3} / \mathrm{s}\). $$\begin{array}{|l|c|c|c|c|c|c|c|}\hline \text { Electrode Position } & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\\hline \text { Percent Decomposition of } \mathrm{HbO}_{2} & 0.00 & 1.93 & 3.82 & 5.68 & 7.48 & 9.25 & 11.00 \\ \hline\end{array}$$ (a) Using the method of differential analysis of rate data, determine the reaction order and the forward specific reaction rate constant \(k\) for the deoxygenation of hemoglobin. (b) Repeat using regression.

Short answer

In summary, we can determine the reaction order and forward specific reaction rate constant (k) for the deoxygenation of hemoglobin using two methods: differential analysis of rate data and regression. First, calculate the reaction rate at each electrode position by finding the change in the percent decomposition of HbO2. Next, for the differential analysis method, plot the reaction rates against the percent decomposition to find the reaction order and use the rate equation to solve for k. For the regression method, create a linear regression model relating the logarithm of reaction rates to the logarithm of the percent decomposition, and use the slope and intercept to find the reaction order and k. Finally, compare the values obtained from both methods to ensure accuracy.

Step-by-step solution

Calculate the reaction rate at each electrode position

To find the reaction rate, we need to calculate the change in the percent decomposition of HbO2 at each electrode position. We can do this by finding the difference between the percent decomposition at two consecutive positions and dividing it by the distance between the electrodes (5 cm). For example, the reaction rate between electrode 1 and 2 can be calculated as follows: Reaction rate = (Percent decomposition at position 2 - Percent decomposition at position 1) / 5 Repeat the same calculation for all consecutive positions.

Perform differential analysis to find reaction order and rate constant k

Using the reaction rates calculated in step 1, we can calculate the reaction order. To do this, we will plot a graph of the reaction rates (y-axis) against the percent decomposition of HbO2 (x-axis). The slope of the graph will give us the reaction order. Once we have the reaction order, we can use it to calculate the rate constant k. We can do this by rearranging the rate equation: Rate = k * [HbO2]^n Where Rate is the reaction rate, k is the rate constant, and n is the reaction order. We can determine k by plugging in the reaction order and the initial rate and percent decomposition values and solving for k.

Calculate reaction order and rate constant k using regression

To find the reaction order and the rate constant k using regression, we will first create a linear regression model that relates the logarithm of the reaction rates to the logarithm of the percent decomposition of HbO2. This can be done using statistical software or a calculator. The slope of this regression line will give us the reaction order (n), and the intercept will give us the logarithm of the rate constant k. We can then exponentiate the intercept to find the actual value of k.

Compare the reaction order and rate constant k obtained by both methods

Finally, we will compare the reaction order and the rate constant k obtained by both methods (differential analysis and regression) to see if they yield similar results. If the results are similar, this will give us confidence in our findings. If not, we might need to re-evaluate our calculations and assumptions.

Key concepts

Deoxygenation kinetics

Deoxygenation kinetics involves the study of how fast oxygen is removed from a substance. In the context of hemoglobin (HbO2) in blood, this means the rate at which hemoglobin loses oxygen. When oxygenated blood passes through tissue capillaries, it releases oxygen to the surrounding tissues, a process described by certain kinetic parameters. Understanding these parameters, such as reaction rate and reaction order, helps in modeling how efficiently hemoglobin delivers oxygen to tissues.

In chemical reaction engineering, kinetics is crucial for developing insights into reaction mechanisms and optimizing conditions for desired outcomes. The goal is to derive a mathematical expression that can predict the behavior of the reaction under different conditions. This requires analyzing experimental data to determine the rate and order of the reaction.

Tubular reactor analysis

A tubular reactor is a type of chemical reactor used to carry out reactions on fluids in motion, such as the deoxygenation process of hemoglobin. This type of reactor is particularly useful for studying reactions that occur continuously along the flow path of the fluid.

In the study by Nakamura and Staub, a tubular reactor was employed to mimic the physiological conditions of blood flow in capillaries. Oxygen electrodes placed at fixed intervals along the tube allowed for measurement of the oxygen concentration changes as the blood moved through the reactor. These measurements were critical for analyzing the reaction kinetics of hemoglobin deoxygenation.

By understanding the tubular reactor setup, students can appreciate how various factors, such as flow rate and reactor dimensions, influence the observed reaction kinetics. It aids in understanding how the spatial distribution of reactants and products varies with time and position in the reactor.

Reaction rate determination

Determining the reaction rate is essential for understanding the kinetics of a chemical process. The reaction rate is the speed at which the reactants are transformed into products. In deoxygenation of hemoglobin, it’s the rate at which HbO2 decomposes to hemoglobin and oxygen.

For the tubular reactor, engineers measure the change in oxygen levels at successive electrode positions. This change over the fixed distance between electrodes allows calculation of the reaction rate as fraction per time and space unit.

• Calculate the difference in percent decomposition between successive electrodes.
• Divide this difference by the distance between electrodes to obtain the rate.

These calculations form the basis for more complex analysis, such as determining the reaction order.

Differential analysis method

The differential analysis method helps identify the reaction order and rate constant by examining how reaction rates change with concentration. This method relies on plotting observed reaction rates against concentrations of reactants, producing a graph whose slope indicates the reaction order.

In our case, the graph is plotted with the reaction rates on the y-axis and percent decomposition of HbO2 on the x-axis. The reaction order, derived from the slope, is crucial for defining the mathematical expression of the reaction rate.

• If the slope is linear, the reaction follows a first-order kinetics.
• Non-linear slopes indicate different orders, requiring further analysis.

After determining the order, you can rearrange the rate equation to solve for the rate constant, using initial conditions from the highest resolution data available.

Regression analysis

Regression analysis is a statistical tool used to identify relationships between variables. In the context of chemical reaction engineering, it is used to determine reaction parameters with high precision.

For the deoxygenation of hemoglobin, regression involves fitting a linear model to the logarithms of reaction rates and concentrations (percent decomposition of HbO2). The mathematical transformation linearizes the data, making it easier to determine reaction order, derived from the slope of the line, and rate constant, derived from the y-intercept.

• Use statistical software to perform regression analysis on logged data.
• The slope provides the reaction order.
• The intercept, when exponentiated, gives the rate constant.

The advantage of regression analysis is its ability to account for variations in experimental data, providing a more robust estimation of reaction parameters compared to simpler methods.