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Chapter 17: Problem 29

(a) What is the ratio of \(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) in blood of pH 7.4 ? (b) What is the ratio of \(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) in an exhausted marathon runner whose blood pH is 7.1\(?\)

Short Answer

Expert verified
(a) In blood with a pH of 7.4, the ratio of \(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) is approximately 20:1. (b) In the blood of an exhausted marathon runner with a pH of 7.1, the ratio of \(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) is 10:1.

Step by step solution

01

Understand the Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is a useful tool in understanding the relationship between pH, pOH, and the concentrations of acids and their conjugate bases: \[ pH = pK_{a} + log_{10}(\frac{[\mathrm{Base}]}{[\mathrm{Acid}]}) \] where \(pH\) is the pH of the buffer solution, \(pK_{a}\) is the negative logarithm of the acid dissociation constant (\(K_{a}\)), and \([\mathrm{Base}]/[\mathrm{Acid}]\) is the ratio of the conjugate base (in our case, \(\mathrm{HCO}_{3}^{-}\)) to the acid (in our case, \(\mathrm{H}_{2} \mathrm{CO}_{3}\)).
02

Determine the pKa value

For the carbonic acid/bicarbonate buffer system, the pKa value is approximately 6.1. We will use this value in our calculations.
03

Calculate the ratio of HCO3- to H2CO3 in blood of pH 7.4 (Part a)

Given the pH of 7.4, we can substitute the known values into the Henderson-Hasselbalch equation and solve for the ratio: \[ 7.4 = 6.1 + log_{10}(\frac{[\mathrm{HCO}_{3}^{-}]}{[\mathrm{H}_{2}\mathrm{CO}_{3}]}) \] First, isolate the logarithm term by subtracting the \(pK_{a}\) value from the pH value: \[ 1.3 = log_{10}(\frac{[\mathrm{HCO}_{3}^{-}]}{[\mathrm{H}_{2}\mathrm{CO}_{3}]}) \] Next, remove the logarithm by taking the antilog (base 10) of both sides: \[ 10^{1.3} = \frac{[\mathrm{HCO}_{3}^{-}]}{[\mathrm{H}_{2}\mathrm{CO}_{3}]} \]
04

Calculate the ratio of HCO3- to H2CO3 in blood of pH 7.1 (Part b)

Repeat the process for the given pH of 7.1: \[ 7.1 = 6.1 + log_{10}(\frac{[\mathrm{HCO}_{3}^{-}]}{[\mathrm{H}_{2}\mathrm{CO}_{3}]}) \] Subtract the \(pK_{a}\) value from the pH value: \[ 1.0 = log_{10}(\frac{[\mathrm{HCO}_{3}^{-}]}{[\mathrm{H}_{2}\mathrm{CO}_{3}]}) \] Remove the logarithm by taking the antilog (base 10) of both sides: \[ 10^{1.0} = \frac{[\mathrm{HCO}_{3}^{-}]}{[\mathrm{H}_{2}\mathrm{CO}_{3}]} \]
05

Present the final ratios

(a) In blood with a pH of 7.4, the ratio of \(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) is approximately \(10^{1.3} \approx 19.95\), or approximately 20:1. (b) In the blood of an exhausted marathon runner with a pH of 7.1, the ratio of \(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) is \(10^{1.0} = 10\), or 10:1.

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Most popular questions from this chapter

Assume that 30.0 \(\mathrm{mL}\) of a 0.10 \(\mathrm{M}\) solution of a weak base \(\mathrm{B}\) that accepts one proton is titrated with a 0.10\(M\) solution of the monoprotic strong acid HA. (a) How many moles of HA have been added at the equivalence point? (b) What is the predominant form of B at the equivalence point? (a) Is the pH \(7,\) less than \(7,\) or more than 7 at the equivalence point?\( (\mathbf{d} )\) Which indicator, phenolphthalein or methyl red, is likely to be the better choice for this titration?

Calculate the \(\mathrm{pH}\) at the equivalence point in titrating 0.100 M solutions of each of the following with 0.080 \(\mathrm{M}\) NaOH: (a) hydrobromic acid (HBr), (b) chlorous acid (HClO_{2} ) , (c) benzoic acid \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COOH}\right)\)

Use values of \(K_{s p}\) for AgI and \(K_{f}\) for \(A g(C N)_{2}^{-}\) to (a) calculate the molar solubility of Agl in pure water, (b) calculate the equilibrium constant for the reaction \(\operatorname{AgI}(s)+2 \mathrm{CN}^{-}(a q) \rightleftharpoons \mathrm{Ag}(\mathrm{CN})_{2}^{-}(a q)+\mathrm{I}^{-}(a q), \quad(\mathbf{c})\) determine the molar solubility of AgI in a 0.100 \(\mathrm{MNaCN}\) solution.

Tooth enamel is composed of hydroxyapatite, whose simplest formula is \(\mathrm{Ca}_{5}\left(\mathrm{PO}_{4}\right)_{3} \mathrm{OH}\) , and whose corresponding \(K_{\mathrm{sp}}=6.8 \times 10^{-27}\) . As discussed in the Chemistry and Life box on page \(746,\) fluoride in fluorinated water or in toothpaste reacts with hydroxyapatite to form fluoroapatite, \(\mathrm{Ca}_{5}\left(\mathrm{PO}_{4}\right)_{3} \mathrm{F},\) whose \(K_{s p}=1.0 \times 10^{-60}\) (a) Write the expression for the solubility-constant for hydroxyapatite and for fluoroapatite. (b) Calculate the molar solubility of each of these compounds.

Consider the equilibrium $$\mathrm{B}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{HB}^{+}(a q)+\mathrm{OH}^{-}(a q)$$ Suppose that a salt of \(\mathrm{HB}^{+}(a q)\) is added to a solution of \(\mathrm{B}(a q)\) at equilibrium. (a) Will the equilibrium constant for the reaction increase, decrease, or stay the same? (b) Will the concentration of \(\mathrm{B}(a q)\) increase, decrease, or stay the same? (c) Will the \(\mathrm{pH}\) of the solution increase, decrease, or stay the same?
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