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Chapter 4: Problem 35

A variation of the indicator-dilution method (see preceding problem) is used to measure total blood volume. A known amount of a tracer is injected into the bloodstream and disperses uniformly throughout the circulatory system. A blood sample is then withdrawn, the tracer concentration in the sample is measured, and the measured concentration [which equals (tracer injected)/(total blood volume) if no tracer is lost through blood vessel walls] is used to determine the total blood volume. In one such experiment, \(0.60 \mathrm{cm}^{3}\) of a solution containing \(5.00 \mathrm{mg} / \mathrm{L}\) of a dye is injected into an artery of a grown man. About 10 minutes later, after the tracer has had time to distribute itself uniformly throughout the bloodstream, a blood sample is withdrawn and placed in the sample chamber of a spectrophotometer. A beam of light passes through the chamber, and the spectrophotometer measures the intensity of the transmitted beam and displays the value of the solution absorbance (a quantity that increases with the amount of light absorbed by the sample). The value displayed is 0.18. A calibration curve of absorbance \(A\) versus tracer concentration \(C\) (micrograms dye/liter blood) is a straight line through the origin and the point \((A=0.9, C=3 \mu \mathrm{g} / \mathrm{L}) .\) Estimate the patient's total blood volume from these data.

Short Answer

Expert verified
The estimated total blood volume in the patient's body is \(0.0556\,\mathrm{L}\) or approximately \(56\,\mathrm{mL}\).

Step by step solution

01

Calculate the amount of dye injected

Begin by finding the total amount of dye that was injected into the blood. This can be done by multiplying the volume of the solution injected by its concentration in the solution. The concentration of the solution is given as \(5.00\,\mathrm{mg} / \mathrm{L}\), and the volume of the solution injected is \(0.60\,\mathrm{cm}^3\). First, convert the volume of the solution from cubic centimeters to liters, as the concentration is given per liter. There are 1000 cubic centimeters in 1 liter, so \(0.60\,\mathrm{cm}^{3} = 0.60\,\mathrm{cm}^{3} * \frac{1\,\mathrm{L}}{1000\,\mathrm{cm}^3} = 0.0006\,\mathrm{L}\). Now, find the total amount of dye that was injected: \(\text{Dye Injected} = (0.0006\,\mathrm{L}) * (5\,\mathrm{mg} / \mathrm{L}) = 0.003\,\mathrm{mg}\).
02

Relate dye concentration to absorbance

The exercise specifies that a straight line can be drawn on a graph with absorbance \(A\) as the y-axis and dye concentration \(C\) as the x-axis. Furthermore, this straight line passes through the origin and the point \((A=0.9, C=3\,\mu\mathrm{g} / \mathrm{L})\). The slope of this line is thus \(0.9 / 3 = 0.3\,\mathrm{mg / L / absorbance unit}\). An absorbance value of 0.18 is observed in the patient's blood sample. The related dye concentration is thus \(0.18 * 0.3 = 0.054\,\mathrm{mg / L}\).
03

Estimate total blood volume

The concentration of dye in the blood is given by the formula \(\text{Concentration} = \text{Total amount of dye} / \text{Total volume of blood}\). Rearranging this formula to solve for the total volume of blood one gets \(\text{Total volume of blood} = \text{Total amount of dye} / \text{Concentration}\). Substituting the total amount of dye injected (0.003 mg), and the concentration of dye in the bloodstream (0.054 mg / L) gives \(\text{Total volume of blood} = 0.003\,\mathrm{mg} / 0.054\,\mathrm{mg / L} = 0.0556\,\mathrm{L}\).

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Key Concepts

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

Indicator-Dilution Method
The indicator-dilution method is a practical approach to measure blood volume by introducing a known quantity of a tracer (or indicator) into the bloodstream. This tracer disperses evenly throughout the blood, allowing us to analyze samples and learn more about blood properties and volume.
Upon injecting the tracer, it travels through the circulatory system, often using a dye due to its clear interaction with light and easy detection. After allowing sufficient time for uniform distribution of the tracer, typically around 10 minutes, the concentration of the tracer in a withdrawn blood sample is examined.
The concentration found in the blood, assuming no loss through vessel walls, directly relates to the total blood volume by the equation: \[\text{Total Blood Volume} = \frac{\text{Amount of Tracer Injected}}{\text{Measured Concentration in Blood}}\] This formula allows the estimation of blood volume by rearranging for total volume, using known and measurable quantities.
Tracer Concentration
Tracer concentration is fundamental to calculating blood volume using the indicator-dilution method. Before injection, we know the concentration of the tracer in the solution. In this exercise, the concentration given was 5.00 mg/L.
Converting the injected solution into a measurable quantity is crucial. The tracer concentration helps determine how much of the dye is present in each liter of blood.
This concentration is derived from the simple multiplication of the volume of the injected solution and the tracer's strength. In this case, after conversion, it is clarified with precise measurements, like understanding that 0.60 cm鲁 of a solution equals 0.0006 L. Then multiplying by 5.00 mg/L gives us the total dye amount: 0.003 mg.
This amount allows the calculation of tracer concentration in the blood after dispersal, which assists in solving for the total blood volume.
Spectrophotometer Calibration
Calibration is crucial when employing a spectrophotometer to ensure accurate measurement readings. Calibration involves creating a standard curve of known quantities to allow for correct interpretation of unknown sample readings.
This experiment reports a linear relationship between absorbance and concentration, highlighted by a calibration curve passing through the origin and a defined point, \((A=0.9, C=3 \mu g / L)\).
The slope of this line, calculated as 0.3 mg/L/absorbance unit, is integral because it translates the absorbance reading produced by the spectrophotometer into the concentration of dye. This relationship requires calibration first, to ensure the device reads correctly under experiment conditions.
Through this calibration, any absorbance value can be confidently converted back to a concentration using the slope, providing clear and precise experimental data.
Absorbance and Concentration Relationship
The relationship between absorbance and concentration in solutions is directly described by Beer's Law, which states that absorbance ( A ) is proportional to concentration ( C ). This characteristic linear relationship makes spectrophotometry incredibly useful in analyzing the concentration of substances.
If a sample has higher absorbance, it indicates more light absorption due to higher concentrations of the tracer dye in this context.
Furthermore, the linearity of the calibration curve allows us to establish a specific mathematical slope, as mentioned before. If a blood sample shows an absorbance of 0.18, a precise concentration can be calculated by multiplying with the slope, 0.3 mg/L per absorbance unit, leading to a concentration of 0.054 mg/L dye.
Through this relationship, absorbance readings can be converted into meaningful data that ultimately allow for the determination of the total volume of blood in the method applied. This conversion is a testament to the effectiveness of applying physical laws in biochemistry.

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Most popular questions from this chapter

Ethane is chlorinated in a continuous reactor: $$\mathrm{C}_{2} \mathrm{H}_{6}+\mathrm{Cl}_{2} \rightarrow \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Cl}+\mathrm{HCl}$$ Some of the product monochloroethane is further chlorinated in an undesired side reaction: $$\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Cl}+\mathrm{Cl}_{2} \rightarrow \mathrm{C}_{2} \mathrm{H}_{4} \mathrm{Cl}_{2}+\mathrm{HCl}$$ (a) Suppose your principal objective is to maximize the selectivity of monochloroethane production relative to dichloroethane production. Would you design the reactor for a high or low conversion of ethane? Explain your answer. (Hint: If the reactor contents remained in the reactor long enough for most of the ethane in the feed to be consumed, what would the main product constituent probably be?) What additional processing steps would almost certainly be carried out to make the process economically sound? (b) Take a basis of \(100 \mathrm{mol} \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Cl}\) produced. Assume that the feed contains only ethane and chlorine and that all of the chlorine is consumed, and carry out a degree-of-freedom analysis based on atomic species balances. (c) The reactor is designed to yield a \(15 \%\) conversion of ethane and a selectivity of \(14 \mathrm{mol} \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Cl} / \mathrm{mol}\) \(\mathrm{C}_{2} \mathrm{H}_{4} \mathrm{Cl}_{2},\) with a negligible amount of chlorine in the product gas. Calculate the feed ratio \(\left(\mathrm{mol} \mathrm{Cl}_{2} /\right.\) mol \(\mathrm{C}_{2} \mathrm{H}_{6}\) ) and the fractional yield of monochloroethane. (d) Suppose the reactor is built and started up and the conversion is only \(14 \% .\) Chromatographic analysis shows that there is no \(\mathrm{Cl}_{2}\) in the product but another species with a molecular weight higher than that of dichloroethane is present. Offer a likely explanation for these results.

Two aqueous sulfuric acid solutions containing \(20.0 \mathrm{wt} \% \mathrm{H}_{2} \mathrm{SO}_{4}(\mathrm{SG}=1.139)\) and \(60.0 \mathrm{wt} \% \mathrm{H}_{2} \mathrm{SO}_{4}\) (SG = 1.498) are mixed to form a 4.00 molar solution (SG = 1.213). (a) Calculate the mass fraction of sulfuric acid in the product solution. (b) Taking \(100 \mathrm{kg}\) of the \(20 \%\) feed solution as a basis, draw and label a flowchart of this process, labeling both masses and volumes, and do the degree-of-freedom analysis. Calculate the feed ratio (liters 20\% solution/liter 60\% solution). (c) What feed rate of the \(60 \%\) solution (L/h) would be required to produce \(1250 \mathrm{kg} / \mathrm{h}\) of the product?

The respiratory process involves hemoglobin (Hgb), an iron-containing compound found in red bloodcells. In the process, carbon dioxide diffuses from tissue cells as molecular \(\mathrm{CO}_{2}\), while \(\mathrm{O}_{2}\) simultaneously enters the tissue cells. A significant fraction of the \(\mathrm{CO}_{2}\) leaving the tissue cells enters red blood cells and reacts with hemoglobin; the \(\mathrm{CO}_{2}\) that does not enter the red blood cells ( \((\mathrm{D}\) in the figure below) remains dissolved in the blood and is transported to the lungs. Some of the \(\mathrm{CO}_{2}\) entering the red blood cells reacts with hemoglobin to form a compound (Hgb. \(\mathrm{CO}_{2} ;(\) 2) in the figure). When the red blood cells reach the lungs, the Hgb.CO_ dissociates, releasing free CO_ Meanwhile, the CO_ that enters the red blood cells but does not react with hemoglobin combines with water to form carbonic acid, \(\mathrm{H}_{2} \mathrm{CO}_{3},\) which then dissociates into hydrogen ions and bicarbonate ions ( (3) in the figure). The bicarbonate ions diffuse out of the cells ( (4) in the figure), and the ions are transported to the lungs via the bloodstream. For adult humans, every deciliter of blood transports a total of \(1.6 \times 10^{-4}\) mol of carbon dioxide in its various forms (dissolved \(\mathrm{CO}_{2}, \mathrm{Hgb} \cdot \mathrm{CO}_{2},\) and bicarbonate ions) from tissues to the lungs under normal, resting conditions. Of the total \(\mathrm{CO}_{2}, 1.1 \times 10^{-4}\) mol are transported as bicarbonate ions. In a typical resting adult human, the heart pumps approximately 5 liters of blood per minute. You have been asked to determine how many moles of \(\mathrm{CO}_{2}\) are dissolved in blood and how many moles of \(\mathrm{Hgb} \cdot \mathrm{CO}_{2}\) are transported to the lungs during an hour's worth of breathing. (a) Draw and fully label a flowchart and do a degree-of-freedom analysis. Write the chemical reactions that occur, and generate, but do not solve, a set of independent equations relating the unknown variables on the flowchart. (b) If you have enough information to obtain a unique numerical solution, do so. If you do not have enough information, identify a specific piece/pieces of information that (if known) would allow you to solve the problem, and show that you could solve the problem if that information were known. (c) When someone loses a great deal of blood due to an injury, they "go into shock": their total blood volume is low, and carbon dioxide is not efficiently transported away from tissues. The carbon dioxide reacts with water in the tissue cells to produce very high concentrations of carbonic acid, some of which can dissociate (as shown in this problem) to produce high levels of hydrogen ions. What is the likely effect of this occurrence on the blood pH near the tissue and the tissue cells? How is this likely to affect the injured person?

Fermentation of sugars obtained from hydrolysis of starch or cellulosic biomass is an alternative to using petrochemicals as the feedstock in production of ethanol. One of the many commercial processes to do this \(^{16}\) uses an enzyme to hydrolyze starch in corn to maltose (a disaccharide consisting of two glucose units) and oligomers consisting of several glucose units. A yeast culture then converts the maltose to ethyl alcohol and carbon dioxide: $$\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}+\mathrm{H}_{2} \mathrm{O}(+\text { yeast }) \rightarrow 4 \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}+4 \mathrm{CO}_{2}\left(+\text { yeast }+\mathrm{H}_{2} \mathrm{O}\right)$$ As the yeast grows, \(0.0794 \mathrm{kg}\) of yeast is produced for every \(\mathrm{kg}\) ethyl alcohol formed, and \(0.291 \mathrm{kg}\) water is produced for every kg of yeast formed. For use as a fuel, the product from such a process must be around 99.5 wt\% ethyl alcohol. Corn fed to the process is 72.0 wt\% starch on a moisture-free basis and contains 15.5 wt\% moisture. It is estimated that 101.2 bushels of corn can be harvested from an acre of com, that each bushel is equivalent to \(25.4 \mathrm{lb}_{\mathrm{m}}\) of corn, and that \(6.7 \mathrm{kg}\) of ethanol can be obtained from a bushel of corn. What acreage of farmland is required to produce 100,000 kg of ethanol product? What factors (economic and environmental) must be considered in comparing production of ethanol by this route with other routes involving petrochemical feedstocks?

Effluents from metal-finishing plants have the potential of discharging undesirable quantities of metals, such as cadmium, nickel, lead, manganese, and chromium, in forms that are detrimental to water and air quality. A local metal-finishing plant has identified a wastewater stream that contains 5.15 wt\% chromium (Cr) and devised the following approach to lowering risk and recovering the valuable metal. The wastewater stream is fed to a treatment unit that removes \(95 \%\) of the chromium in the feed and recycles it to the plant. The residual liquid stream leaving the treatment unit is sent to a waste lagoon. The treatment unit has a maximum capacity of 4500 kg wastewater/h. If wastewater leaves the finishing plant at a rate higher than the capacity of the treatment unit, the excess (anything above \(4500 \mathrm{kg} / \mathrm{h}\) ) bypasses the unit and combines with the residual liquid leaving the unit, and the combined stream goes to the waste lagoon. (a) Without assuming a basis of calculation, draw and label a flowchart of the process. (b) Wastewater leaves the finishing plant at a rate \(\dot{m}_{1}=6000 \mathrm{kg} / \mathrm{h}\). Calculate the flow rate of liquid to the waste lagoon, \(\dot{m}_{6}(\mathrm{kg} / \mathrm{h}),\) and the mass fraction of \(\mathrm{Cr}\) in this liquid, \(x_{6}(\mathrm{kg} \mathrm{Cr} / \mathrm{kg})\) (c) Calculate the flow rate of liquid to the waste lagoon and the mass fraction of Crin this liquid for \(\dot{m}_{1}\) varying from \(1000 \mathrm{kg} / \mathrm{h}\) to \(10,000 \mathrm{kg} / \mathrm{h}\) in \(1000 \mathrm{kg} / \mathrm{h}\) increments. Generate a plot of \(x_{6}\) versus \(\dot{m}_{1}\). (Suggestion: Use a spreadsheet for these calculations.) (d) The company has hired you as a consultant to help them determine whether or not to add capacity to the treatment unit to increase the recovery of chromium. What would you need to know to make this determination? (e) What concerns might need to be addressed regarding the waste lagoon?
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