Question

Following a bout of intense exercise, the \(\mathrm{pH}\) of the exerciser's blood was found to be \(7.1 .\) If the \(\mathrm{HCO}_{3}^{-}\) concentration is \(8 \mathrm{mM},\) and the \(\mathrm{p} K_{\mathrm{a}}\) for \(\mathrm{HCO}_{3}^{-}\) is \(6.1,\) what is the concentration of \(\mathrm{CO}_{2}\) in the blood? \({6}\)

Short answer

The concentration of \(\text{CO}_2\) in the blood is \(0.8 \ \text{mM}\).

Step-by-step solution

Understanding the Problem

We need to find the concentration of \(\text{CO}_2\) in the blood given the \(\text{pH}\), \([\text{HCO}_3^-]\), and \(\text{p}K_a\). We will use the Henderson-Hasselbalch equation for this balance.

Recall the Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is given by: \[ \text{pH} = \text{p}K_a + \log \left( \frac{[\text{HCO}_3^-]}{[\text{CO}_2]} \right) \] This equation relates the pH of the blood to the concentrations of \(\text{HCO}_3^-\) and \(\text{CO}_2\).

Substitute Known Values

Substitute \(\text{pH} = 7.1\), \(\text{p}K_a = 6.1\), and \([\text{HCO}_3^-] = 8 \ \text{mM}\) into the equation. \[ 7.1 = 6.1 + \log \left( \frac{8}{[\text{CO}_2]} \right) \]

Solve for Logarithm

Rearrange the equation to isolate the logarithm. \[ \log \left( \frac{8}{[\text{CO}_2]} \right) = 7.1 - 6.1 \] \[ \log \left( \frac{8}{[\text{CO}_2]} \right) = 1 \]

Solve for Concentration of CO2

To remove the logarithm, rewrite the equation in exponential form: \[ \frac{8}{[\text{CO}_2]} = 10^1 \] Simplify to find \([\text{CO}_2]\): \[ [\text{CO}_2] = \frac{8}{10} = 0.8 \ \text{mM} \]

Conclusion

The concentration of \(\text{CO}_2\) in the blood is \(0.8 \ \text{mM}\).

Key concepts

Acid-Base Balance

Acid-base balance is essential for maintaining the body's internal environment. When we speak of acid-base balance, we're talking about the mechanisms that the body uses to maintain a stable pH level. This balance is crucial because enzymes, cellular functions, and metabolic processes work optimally only within a narrow pH range.
The bicarbonate ( [HCO_3^-] ) and carbon dioxide ( [CO_2] ) levels in the blood play a significant role in maintaining this balance. The presence of these molecules allows the body to buffer acids and bases, controlling the pH.
During intense exercise, for instance, the production of lactic acid can lower blood pH. The buffet capacity of bicarbonate helps neutralize this acid, illustrating the body's dynamic system of maintaining acid-base homeostasis.

pH Calculation

Calculating pH is a fundamental concept in chemistry, especially in biological contexts. The pH scale is logarithmic, indicating the hydrogen ion concentration in a solution. A pH of 7 is considered neutral, whereas a pH below 7 is acidic, and above 7 is basic.
For blood, a stable pH level (typically around 7.4) is critical for health. The calculation of pH can be achieved using the Henderson-Hasselbalch equation when dealing with buffer solutions like blood.
In our exercise, you can see how the pH of 7.1, more acidic than usual, is used in conjunction with the [HCO_3^-] concentration and the pK_a value to find the [CO_2] concentration using the equation. Understanding this tool is vital for interpreting changes in blood chemistry.

Blood Chemistry

Blood chemistry encompasses the study of chemical constituents within the blood that are critical to maintaining physiological function. A key aspect of blood chemistry is the analysis of gases like oxygen and carbon dioxide, ions like sodium and potassium, and metabolites like glucose and cholesterol.
Specifically, bicarbonate and carbon dioxide are crucial in maintaining the acid-base balance in our bodies as seen in our exercise problem. They also provide insights into respiratory and metabolic status. For example, variations in [HCO_3^-] and [CO_2] can help assess conditions like metabolic acidosis or respiratory alkalosis.
Labs will measure these levels to diagnose diseases, examine organ function, and monitor health status. Thus, understanding blood chemistry through problems like this provides foundational knowledge for many diagnostic processes.

Biochemistry Education

Biochemistry education involves learning about the chemical processes within and related to living organisms. It integrates chemistry and biology and is an essential discipline in understanding the molecular foundations of life.
Educational exercises like the one we've discussed are valuable in teaching students how to evaluate physiological processes such as acid-base balance using mathematical models and equations.
By engaging with textbook exercises and understanding solutions within biochemistry, students develop problem-solving skills and an appreciation for the complexity of biochemical reactions. These exercises not only build foundational knowledge but also prepare students for advanced courses in medicine, pharmacology, and related fields.