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Chapter 17: Problem 114

What is the \(\mathrm{pH}\) at \(25^{\circ} \mathrm{C}\) of water saturated with \(\mathrm{CO}_{2}\) at a partial pressure of \(111.5 \mathrm{kPa}\) ? The Henry's law constant for \(\mathrm{CO}_{2}\) at \(25^{\circ} \mathrm{C}\) is \(3.1 \times 10^{-4} \mathrm{~mol} / \mathrm{L}-\mathrm{kPa}\).

Short Answer

Expert verified
The pH of water saturated with CO鈧� at a partial pressure of 111.5 kPa and a temperature of 25掳C is approximately 4.44.

Step by step solution

01

Determine the dissolved CO鈧� concentration using Henry's Law constant

Using Henry's Law constant, we can calculate the concentration of CO鈧� dissolved in water: \[ C_{CO_{2}} = k_{H} \times P \] where \(C_{CO_{2}}\) is the concentration of CO鈧�, \(k_{H}\) is Henry's Law constant, and \(P\) is the partial pressure of CO鈧�. \[ C_{CO_{2}} = (3.1 \times 10^{-4} \, \mathrm{mol/L/kPa}) \times (111.5 \, \mathrm{kPa}) \] \[ C_{CO_{2}} = 0.034565 \, \mathrm{mol/L} \]
02

Write the dissolution and formation equation of carbonic acid

The dissolution of CO鈧� in water and the subsequent formation of carbonic acid (H鈧侰O鈧�) can be represented as follows: \[ CO_{2} \, (g) + H_{2}O \, (l) \rightleftharpoons H_{2}CO_{3} \, (aq) \]
03

Write the ionization equation of carbonic acid

The ionization of carbonic acid (H鈧侰O鈧�) into hydrogen ions (H鈦�) and bicarbonate ions (HCO鈧冣伝) can be represented as follows: \[ H_{2}CO_{3} \, (aq) \rightleftharpoons H^{+} \, (aq) + HCO_{3}^{-} \, (aq) \]
04

Calculate the hydrogen ion concentration using the equilibrium constant

The equilibrium constant (Ka1) for the ionization of carbonic acid is \(4.45 \times 10^{-7}\). Using this equilibrium constant, we can calculate the concentration of hydrogen ions (H鈦�): \[ K_{a1} = \frac{[H^{+}][HCO_{3}^{-}]}{[H_{2}CO_{3}]} \] Assuming that the concentration of H鈦� and HCO鈧冣伝 ions is the same (since they both come from the ionization of one molecule of H鈧侰O鈧�) and using the concentration of H鈧侰O鈧� obtained in step 1, we can solve for the H鈦� concentration: \[ 4.45 \times 10^{-7} = \frac{[H^{+}]^{2}}{0.034565} \] \[ [H^{+}]^{2} = (4.45 \times 10^{-7}) \times 0.034565 \] \[ [H^{+}] = \sqrt{(4.45 \times 10^{-7}) \times 0.034565} = 3.59 \times 10^{-5} \, \mathrm{mol/L} \]
05

Calculate the pH

Now that we have the concentration of hydrogen ions (H鈦�), we can calculate the pH using the following equation: \[ pH = -\log{[H^{+}]} \] \[ pH = -\log{(3.59 \times 10^{-5})} = 4.44 \] So, the pH of water saturated with CO鈧� at a partial pressure of 111.5 kPa and a temperature of 25掳C is approximately 4.44.

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Key Concepts

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

Henry's Law
When a gas is in contact with a liquid, it can dissolve into the liquid. The amount of gas that dissolves is directly proportional to its partial pressure above the liquid. This concept is defined by Henry's Law. For a particular gas, we express this with the formula:
  • \[ C_{gas} = k_H \times P \]
Here, \(C_{gas}\) is the concentration of gas in the liquid, \(k_H\) is Henry's Law constant, and \(P\) is the partial pressure of the gas.
This principle tells us that a higher pressure results in more gas being dissolved. In the context of our exercise, this means knowing the partial pressure of carbon dioxide \(CO_2\) allows us to calculate how much of it is dissolved in water.
Carbonic Acid
Carbonic acid (\(H_2CO_3\)) is a weak acid that forms when carbon dioxide (\(CO_2\)) dissolves in water. The process can be described by the chemical reaction:
  • \[ CO_2 (g) + H_2O (l) \rightleftharpoons H_2CO_3 (aq) \]
This equilibrium shows that carbonic acid exists naturally as a result of \(CO_2\) in the atmosphere interacting with rainwater. Carbonic acid plays a key role in maintaining the pH balance in aqueous solutions and in natural systems like our blood and the oceans.
Understanding this reaction is essential as it sets the stage for further reactions when assessing the pH levels using carbonate compounds.
pH Calculation
The pH of a solution is a measure of how acidic or basic it is. Specifically, it is the negative logarithm of the hydrogen ion concentration:
  • \[ pH = -\log{[H^+]} \]
In our specific exercise, we used Henry's Law to find the concentration of dissolved \(CO_2\), which then allowed us to calculate the concentration of hydrogen ions using the ionization of carbonic acid. Calculating the pH involves substituting this hydrogen ion concentration into the pH formula above.
This calculation provides insight into the acidity of rainwater or the ocean, both influenced by \(CO_2\) absorption.
Equilibrium Constant
The equilibrium constant (\(K_a\)) expresses the extent of dissociation or ionization of a solute in a solution. For carbonic acid, the equilibrium constant for its first ionization is significant:
  • \[ K_{a1} = \frac{[H^+][HCO_3^-]}{[H_2CO_3]} \]
A higher \(K_a\) value indicates a stronger acid that dissociates more in solution. In our exercise, the given \(K_{a1}\) for \(H_2CO_3\) allowed us to find the \([H^+]\) concentration from the initial concentration of carbonic acid.
This process illustrates the relationship between reactants and products in a reversible chemical reaction, providing a quantitative measure of their equilibrium.

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Most popular questions from this chapter

A buffer, consisting of \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\) and \(\mathrm{HPO}_{4}^{2-},\) helps control the pH of physiological fluids. Many carbonated soft drinks also use this buffer system. What is the \(\mathrm{pH}\) of a soft drink in which the major buffer ingredients are \(10.0 \mathrm{~g}\) of \(\mathrm{KH}_{2} \mathrm{PO}_{4}\) and \(10.0 \mathrm{~g}\) of \(\mathrm{K}_{2} \mathrm{HPO}_{4}\) per \(0.500 \mathrm{~L}\) of solution?

(a) True or false: "solubility" and "solubility-product constant" are the same number for a given compound. (b) Write the expression for the solubility- product constant for each of the following ionic compounds: \(\mathrm{MnCO}_{3}, \mathrm{Hg}(\mathrm{OH})_{2},\) and \(\mathrm{Cu}_{3}\left(\mathrm{PO}_{4}\right)_{2}\).

What is the \(\mathrm{pH}\) of a solution made by mixing \(0.40 \mathrm{~mol}\) \(\mathrm{NaOH}, 0.25 \mathrm{~mol} \mathrm{Na}_{2} \mathrm{HPO}_{4}\), and \(0.30 \mathrm{~mol} \mathrm{H}_{3} \mathrm{PO}_{4}\) with water and diluting to \(2.00 \mathrm{~L} ?\)

A sample of \(500 \mathrm{mg}\) of an unknown monoprotic acid was dissolved in \(50.0 \mathrm{~mL}\) of water and titrated with \(0.200 \mathrm{M}\) KOH. The acid required \(20.60 \mathrm{~mL}\) of base to reach the equivalence point. (a) What is the molar mass of the acid? (b) After \(10.30 \mathrm{~mL}\) of base had been added in the titration, the pH was found to be 4.20 . What is the \(\mathrm{p} K_{a}\) for the unknown acid?

A solution contains \(1.0 \times 10^{-4} \mathrm{Ca}^{2+}(a q)\) and \(1.0 \times 10^{-4}\) \(\mathrm{La}^{3+}(a q) .\) If \(\mathrm{NaF}\) is added, will \(\mathrm{CaF}_{2}\left(K_{s p}=3.9 \times 10^{-11}\right)\) or \(\mathrm{LaF}_{3}\left(K_{s p}=2 \times 10^{-19}\right)\) precipitate first? Specify the concentration of \(\mathrm{F}^{-}(a q)\) needed to begin precipitation.
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