/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 77 The ratio of \(\mathrm{HCO}_{3}^... [FREE SOLUTION] | 91Ó°ÊÓ

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The ratio of \(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) in blood is called the "bicarb number" and is used as a measure of blood \(\mathrm{pH}\) in hospital emergency rooms. A newly diagnosed diabetic patient is admitted to the emergency room with ketoacidosis and a bicarb number of \(10 .\) Calculate the blood \(\mathrm{pH} . K_{\mathrm{a}}\) for carbonic acid at body temperature \(\left(37^{\circ} \mathrm{C}\right)\) is \(7.9 \times 10^{-7}\)

Short Answer

Expert verified
The blood pH is 7.1.

Step by step solution

01

Understand the problem

We need to find the pH of the blood given the ratio (bicarb number) of \( \mathrm{HCO}_3^- \) to \( \mathrm{H}_2\mathrm{CO}_3 \) and the \( K_a \) of carbonic acid.
02

Apply the Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is used to relate the ratio of the concentrations of the ionized form and the non-ionized form of an acid to the pH. It is given by: \[ \text{pH} = \text{pK}_a + \log \left( \frac{\left[\mathrm{HCO}_3^-\right]}{\left[\mathrm{H}_2\mathrm{CO}_3\right]} \right) \].
03

Calculate the pKa from Ka

First, calculate the \( \text{pK}_a \) from \( K_a \). Use the formula: \( \text{pK}_a = -\log( K_a ) \). Here, \( K_a = 7.9 \times 10^{-7} \).
04

Compute the pKa

Calculate \( \text{pK}_a \) using the expression: \[ \text{pK}_a = -\log(7.9 \times 10^{-7}) \]. This results in \( \text{pK}_a \approx 6.1 \).
05

Substitute values into the equation

Now substitute the \( \text{pK}_a \) and the bicarb number into the Henderson-Hasselbalch equation: \[ \text{pH} = 6.1 + \log(10) \].
06

Calculate the pH

Compute \( \log(10) \), which is 1, and complete the equation: \[ \text{pH} = 6.1 + 1 \] resulting in \( \text{pH} = 7.1 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Henderson-Hasselbalch equation
The Henderson-Hasselbalch equation is a crucial tool for calculating the pH of a solution, especially when dealing with weak acids and their conjugate bases. It simplifies the relationship between the pH of a solution, the pKa of the acid, and the concentrations of the acid and its conjugate base. This equation is represented as:\[\text{pH} = \text{pK}_a + \log \left( \frac{[\text{Base}]}{[\text{Acid}]} \right)\]In the context of blood chemistry, the equation is used to ascertain the blood pH by considering the ratio of bicarbonate ions (\(\mathrm{HCO}_3^-\)) to carbonic acid (\(\mathrm{H}_2\mathrm{CO}_3\)). The pH level indicates whether the blood is more acidic or alkaline. Given the bicarb number, which is this ratio, medical professionals can quickly assess a patient's blood pH. It’s particularly useful in emergency situations like ketoacidosis, where time is of the essence, as it allows for rapid intervention to stabilize blood pH levels.
Carbonic acid
Carbonic acid plays a pivotal role in maintaining the acid-base balance in our blood. This weak acid forms when carbon dioxide dissolves in water, which is a common occurrence in our bodily functions.The dissociation of carbonic acid in blood is a key reaction: \[\mathrm{H}_2\mathrm{CO}_3 \rightleftharpoons \mathrm{H}^+ + \mathrm{HCO}_3^-\]In the bloodstream, this dissociation partly controls the concentration of hydrogen ions, thereby influencing the blood pH. Carbonic acid is an essential component of the bicarbonate buffer system, which ensures our blood remains within a tight pH range crucial for physiological processes.
Disturbances in levels of carbonic acid or bicarbonate ions can lead to health issues, emphasizing the importance of this component in not only maintaining pH balance but also in clinical diagnostics, especially for conditions like metabolic acidosis. Medical professionals often measure the ratio of bicarbonate to carbonic acid when assessing changes in blood pH.
Ketoacidosis
Ketoacidosis, particularly diabetic ketoacidosis (DKA), is a serious condition that arises when the body produces excess ketones, leading to an acidic blood environment. This is often seen in people with uncontrolled diabetes. During ketoacidosis, blood pH can drop to dangerously low levels as the accumulation of ketones increases acidity.
Symptoms of ketoacidosis include:
  • Frequent urination
  • Extreme thirst
  • High blood sugar levels
  • Nausea and vomiting
  • Fruity-scented breath
It occurs when there is not enough insulin to help the cells use glucose for energy, prompting the body to break down fat as an alternative energy source.
This breakdown process produces ketones, which are harmful in large amounts.
The body's buffer systems, such as the bicarbonate buffering system connected to carbonic acid, struggle to maintain equilibrium, further complicating conditions. In medical settings, the Henderson-Hasselbalch equation and the bicarb number are used to evaluate the severity and guide treatment of ketoacidosis. Immediate treatment often involves insulin therapy and rehydration to restore normal pH balance.

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Most popular questions from this chapter

Determine whether \(\mathrm{Cd}^{2+}\) can be separated from \(\mathrm{Zn}^{2+}\) by bubbling \(\mathrm{H}_{2} \mathrm{~S}\) through a \(0.3 \mathrm{M} \mathrm{HCl}\) solution that contains \(0.005 \mathrm{M} \mathrm{Cd}^{2+}\) and \(0.005 \mathrm{M} \mathrm{Zn}^{2+} .\left(K_{\mathrm{spa}}\right.\) for CdS is \(8 \times 10^{-7} .\) )

For each of the following, write the equilibrium-constant expression for \(K_{\mathrm{sp}}\) : (a) \(\mathrm{Ca}(\mathrm{OH})_{2}\) (b) \(\mathrm{Ag}_{3} \mathrm{PO}_{4}\) (c) \(\mathrm{BaCO}_{3}\) (d) \(\mathrm{Ca}_{5}\left(\mathrm{PO}_{4}\right)_{3} \mathrm{OH}\)

Some progressive hair coloring products marketed to men, such as Grecian Formula 16, contain lead acetate, \(\mathrm{Pb}\left(\mathrm{CH}_{3} \mathrm{CO}_{2}\right)_{2} .\) As the coloring solution is rubbed on the hair, the \(\mathrm{Pb}^{2+}\) ions react with the sulfur atoms in hair proteins to give lead(II) sulfide (PbS), which is black. A typical coloring solution contains \(0.3\) mass \(\% \mathrm{~Pb}\left(\mathrm{CH}_{3} \mathrm{CO}_{2}\right)_{2}\), and about \(2 \mathrm{~mL}\) of the solution is used per application. (a) Assuming that \(30 \%\) of the \(\mathrm{Pb}\left(\mathrm{CH}_{3} \mathrm{CO}_{2}\right)_{2}\) is converted to \(\mathrm{PbS}\), how many milligrams of \(\mathrm{PbS}\) are formed per application of the coloring solution? (b) Suppose the hair is washed with shampoo and water that has \(\mathrm{pH}=5.50 .\) How many washings would be required to remove \(50 \%\) of the black color? Assume that 3 gal of water is used per washing and that the water becomes saturated with \(\mathrm{PbS}\). (c) Does the calculated number of washings look reasonable, given that frequent application of the coloring solution is recommended? What process(es) in addition to dissolution might contribute to the loss of color?

Which of the following compounds are more soluble in acidic solution than in pure water? (a) \(\mathrm{AgCN}\) (b) \(\mathrm{PbI}_{2}\) (c) \(\mathrm{Al}(\mathrm{OH})_{3}\) (d) \(\mathrm{ZnS}\)

Phenol \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{OH}, K_{\mathrm{a}}=1.3 \times 10^{-10}\right)\) is a weak acid used in mouthwashes, and pyridine \(\left(\mathrm{C}_{5} \mathrm{H}_{5} \mathrm{~N}, K_{\mathrm{b}}=1.8 \times 10^{-9}\right)\) is a weak base used as a solvent. Calculate the value of \(K_{\mathrm{n}}\) for the neutralization of phenol by pyridine. Does the neutralization reaction proceed very far toward completion?

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