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The Henry's law constant for \(\mathrm{CO}_{2}\) in water at \(25^{\circ} \mathrm{C}\) is \(3.1 \times 10^{-2} \mathrm{M} \mathrm{atm}^{-1} .\) (a) What is the solubility of \(\mathrm{CO}_{2}\) in water at this temperature if the solution is in contact with air at normal atmospheric pressure? (b) Assume that all of this \(\mathrm{CO}_{2}\) is in the form of \(\mathrm{H}_{2} \mathrm{CO}_{3}\) produced by the reaction between \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O}\) : $$ \mathrm{CO}_{2}(a q)+\mathrm{H}_{2} \mathrm{O}(l)-\longrightarrow \mathrm{H}_{2} \mathrm{CO}_{3}(a q) $$ What is the \(\mathrm{pH}\) of this solution?

Short Answer

Expert verified
The solubility of CO鈧 in water at \(25^\circ \mathrm{C}\) and normal atmospheric pressure is \(1.27\times10^{-5}\: \mathrm{M}\). When all the dissolved CO鈧 is in the form of H鈧侰O鈧, the pH of the solution is 6.02.

Step by step solution

01

Define Henry's Law

Henry's Law can be used to determine the solubility of a gas in a liquid. The law states that the concentration of a dissolved gas in a solution is proportional to the partial pressure of that gas above the solution: \[C = k_H \cdot P\] where \(C\) is the concentration of the dissolved gas (in mol/L), \(k_H\) is the Henry's law constant (in mol/L atm), and \(P\) is the partial pressure of the gas above the solution (in atm).
02

Calculate the solubility of CO鈧 at normal atmospheric pressure

We are given the value of \(k_H\) for CO鈧 at \(25^\circ \mathrm{C}\), which is \(3.1 脳 10^{-2} \mathrm{M \ atm}^{-1}\). The partial pressure of CO鈧 in the air at normal atmospheric pressure can be taken as about 410 ppm or \(4.10 \times 10^{-4}\) atm, as it is the dominant form at this pressure. Using Henry's Law, we can calculate the solubility of CO鈧 in water: \[C = k_H \cdot P\] \[C = (3.1\times10^{-2}\: \mathrm{M} \mathrm{atm}^{-1}) (4.10 \times 10^{-4} \: \mathrm{atm})\] \[C = 1.27\times10^{-5}\: \mathrm{M}\] The solubility of CO鈧 in water at \(25^\circ \mathrm{C}\) and normal atmospheric pressure is \(1.27\times10^{-5}\: \mathrm{M}\).
03

Define the dissociation of carbonic acid and analyze its equilibrium constant

Carbonic acid is a weak acid that partially dissociates in water. The net reaction is represented by: \[\mathrm{H}_2\mathrm{CO}_3(aq) \leftrightarrows \mathrm{H}^+(aq) + \mathrm{HCO}_3^-(aq)\] The equilibrium constant for this reaction, \(K_a\), is: \[K_a = \frac{[\mathrm{H}^+][\mathrm{HCO}_3^-]}{[\mathrm{H}_2\mathrm{CO}_3]}\] For carbonic acid, \(K_a\) is around \(4.45 脳 10^{-7}\).
04

Calculate the pH of the solution

Since we are assuming that all the dissolved CO鈧 is in the form of H鈧侰O鈧, we can write the equilibrium expression: \[\frac{x^2}{1.27\times10^{-5}M - x} = 4.45\times10^{-7}\] Here, \(x\) represents the concentration of H鈦 ions and HCO鈧冣伝 ions. Since the value of \(K_a\) is quite small, we can make the approximation that \(x \ll 1.27\times10^{-5}\: \mathrm{M}\), so the denominator can be approximated as \(1.27\times10^{-5}\: \mathrm{M}\). So, \[\frac{x^2}{1.27\times10^{-5}M} = 4.45\times10^{-7}\] \[x^2 = (1.27\times10^{-5}M)(4.45\times10^{-7})\] \[x = \sqrt{(1.27\times10^{-5})(4.45\times10^{-7})}\] \[x = 9.50\times10^{-7}\: \mathrm{M}\] Now, we can calculate the pH of the solution: \[\mathrm{pH} = -\log_{10}[\mathrm{H}^+]\] \[\mathrm{pH} = -\log_{10}(9.50\times10^{-7})\] \[\mathrm{pH} = 6.02\] The pH of this solution is 6.02.

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Key Concepts

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

Solubility
When discussing solubility, we talk about how much of a substance (like a gas) can dissolve into a liquid until it reaches an equilibrium. This is all about the balance between the liquid absorbing the gas and the gas escaping back to the air. How does this help us? In applications where we want gases to stay dissolved, knowing solubility is crucial.

The formula of solubility given by Henry's Law is essential for calculating this. Henry's Law states that the concentration of a gas dissolved in a solution is directly proportional to the partial pressure of the gas above. This can be expressed mathematically as:
  • \( C = k_H \cdot P \)
Here, \( C \) is the concentration of the dissolved gas, \( k_H \) is Henry's law constant, and \( P \) is the partial pressure of the gas.

For example, with carbon dioxide and water, this law allows us to calculate how much CO鈧 will stay dissolved in water under normal atmospheric pressure. Its solubility at 25掳C is about \(1.27 \times 10^{-5}\: \mathrm{M}\). Understanding this concept helps us in scenarios ranging from environmental science to beverage carbonation.
Carbonic Acid
Carbonic acid is a fascinating part of chemistry, as it's the product of carbon dioxide (CO鈧) dissolving in water. When CO鈧 interacts with water (H鈧侽), it forms carbonic acid (H鈧侰O鈧):
  • \( \mathrm{CO}_2 (aq) + \mathrm{H}_2 \mathrm{O} (l) \rightarrow \mathrm{H}_2 \mathrm{CO}_3 (aq) \)
This reaction illustrates how gases like CO鈧 can become part of liquid solutions, often dramatically altering the properties of the liquid.

In aquifers and oceans, carbonic acid plays a vital role in maintaining environmental pH stability. It acts both by forming and dissociating easily to respond to shifts in pH and maintain equilibrium. This makes it key to processes like buffering, which ensures that an environment doesn't become too acidic or basic.

Understanding the transformation of CO鈧 to H鈧侰O鈧 also illuminates the greater role of carbon cycling in nature, influencing phenomena such as acid rain and atmospheric CO鈧 levels.
Weak Acid Dissociation
Weak acids like carbonic acid (H鈧侰O鈧) do not completely dissociate in water. This partial dissociation creates a dynamic balance between the undissociated molecules and the ions formed. The process is governed by an equilibrium constant known as the acid dissociation constant (\( K_a \)).

For carbonic acid, the dissociation reaction in water can be shown as:
  • \( \mathrm{H}_2\mathrm{CO}_3 (aq) \leftrightarrows \mathrm{H}^+ (aq) + \mathrm{HCO}_3^- (aq) \)
The equilibrium expression for this reaction is:
  • \( K_a = \frac{[\mathrm{H}^+][\mathrm{HCO}_3^-]}{[\mathrm{H}_2\mathrm{CO}_3]} \)
With a small \( K_a \) value of about \(4.45 \times 10^{-7}\), we can assume that the concentrations of ions formed are also low, pointing to its weak nature.

This weak acid behavior influences calculations like the pH of solutions, where only a fraction of H鈧侰O鈧 contributes to the acidity. Understanding this concept allows us to predict and quantify the acidity of solutions, an essential skill in both laboratory and environmental chemistry.

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Most popular questions from this chapter

The hydroxyl radical, \(\mathrm{OH}\), is formed at low altitudes via the reaction of excited oxygen atoms with water: $$ \mathrm{O}^{4}(g)+\mathrm{H}_{2} \mathrm{O}(g) \longrightarrow 2 \mathrm{OH}(g) $$ (a) Write the Lewis structure for the hydroxyl radical (Hint: It has one unpaired electron.) Once produced, the hydroxyl radical is very reactive. Explain why each of the following series of reactions affects the pollution in the troposphere: (b) \(\mathrm{OH}+\mathrm{NO}_{2} \longrightarrow \mathrm{HNO}_{3}\) (c) \(\mathrm{OH}+\mathrm{CO}+\mathrm{O}_{2} \longrightarrow \mathrm{CO}_{2}+\mathrm{OOH}\) \(\mathrm{OOH}+\mathrm{NO} \longrightarrow \mathrm{OH}+\mathrm{NO}_{2}\) (d) \(\mathrm{OH}+\mathrm{CH}_{4} \longrightarrow \mathrm{H}_{2} \mathrm{O}+\mathrm{CH}_{3}\) \(\mathrm{CH}_{3}+\mathrm{O}_{2} \longrightarrow \mathrm{OOCH}_{3}\) \(\mathrm{OOCH}_{3}+\mathrm{NO} \longrightarrow \mathrm{OCH}_{3}+\mathrm{NO}_{2}\)

If an average \(\mathrm{O}_{3}\) molecule "lives" only \(100-200\) seconds in the stratosphere before undergoing dissociation, how can \(\mathrm{O}_{3}\) offer any protection from ultraviolet radiation?

Explain why increasing concentrations of \(\mathrm{CO}_{2}\) in the atmosphere affect the quantity of energy leaving Earth but do not affect the quantity entering from the Sun.

As of the writing of this text, EPA standards limit atmospheric ozone levels in urban environments to 84 ppb. How many moles of ozone would there be in the air above Los Angeles County (area about 4000 square miles; consider a height of \(10 \mathrm{~m}\) above the ground) if ozone was at this concentration?

(a) With respect to absorption of radiant energy, what distinguishes a greenhouse gas from a nongreenhouse gas? (b) \(\mathrm{CH}_{4}\) is a greenhouse gas, but Ar is not. How might the molecular structure of \(\mathrm{CH}_{4}\) explain why it is a greenhouse gas?

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