Problem 119 - Chemistry: The Central Science in SI Units, Global Edition

Question

Atmospheric \(\mathrm{CO}_{2}\) levels have risen by nearly \(20 \%\) over the past 40 years from 320 ppm to 400 ppm. (a) Given that the average \(\mathrm{pH}\) of clean, unpolluted rain today is 5.4 , determine the \(\mathrm{pH}\) of unpolluted rain 40 years ago. Assume that carbonic acid \(\left(\mathrm{H}_{2} \mathrm{CO}_{3}\right)\) formed by the reaction of \(\mathrm{CO}_{2}\) and water is the only factor influencing \(\mathrm{pH}\). $$ \mathrm{CO}_{2}(g)+\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{H}_{2} \mathrm{CO}_{3}(a q) $$ (b) What volume of \(\mathrm{CO}_{2}\) at \(25^{\circ} \mathrm{C}\) and \(101.3 \mathrm{kPa}\) is dissolved in a 20.0-L bucket of today's rainwater?

Step-by-Step Solution

Answer

(a) By calculating the hydrogen ion concentration in today's rainwater, we find that the pH of unpolluted rain 40 years ago was approximately \(5.2\). (b) The volume of CO₂ dissolved in a 20.0-L bucket of today's rainwater at \(25^{\circ} \mathrm{C}\) and \(101.3 \mathrm{kPa}\) is approximately \(7.02 \times 10^{-3} \mathrm{m}^3\).

Step 1: (a) Determine the concentration of carbonic acid in today's rainwater.

To determine the pH 40 years ago, we need to first determine the concentration of carbonic acid in today's rainwater. The relationship between pH and the concentration of ions is given by the formula: \(pH = -\log_{10} [H^{+}]\) where [H⁺] denotes the concentration of hydrogen ions in the rainwater. We know the pH of clean, unpolluted rain today is 5.4. Using this pH value, we can find the concentration of hydrogen ions: \(5.4 = -\log_{10} [H^{+}]\) Now, solve for [H⁺]: \([H^{+}] = 10^{-5.4}\)

Step 2: (a) Determine the original ratio of carbonic acid concentration and CO₂ concentration.

We know that the carbonic acid present in the rainwater is due to the presence of CO₂. So, we need to find the relationship between the concentration of carbonic acid and the concentration of CO₂. According to the balanced equation for this reaction: \(\mathrm{CO}_{2}(g)+\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{H}_{2} \mathrm{CO}_{3}(a q)\) This shows that every one mole of CO₂ dissolves in water to form one mole of carbonic acid. Hence, we can conclude that the ratio of the original concentration of carbonic acid and CO₂ is constant. Let's denote this constant ratio as: \(K = \frac{[\mathrm{H}_{2} \mathrm{CO}_{3}]}{[\mathrm{CO}_{2}]}\)

Step 3: (a) Calculate the CO₂ concentration and pH 40 years ago.

Now, we know that the atmospheric CO₂ levels have increased by 20% over the past 40 years i.e., the concentration of CO₂ has increased from 320 ppm to 400 ppm. So, we can write the CO₂ concentration 40 years ago as (320/400) times today's CO₂ concentration. Let the CO₂ concentration 40 years ago be denoted by [CO₂_original]. Then, \([CO_{2original}] = \frac{320}{400} [CO_{2today}]\) Now, we can use the constant ratio 'K' that we established earlier to find the original concentration of carbonic acid: \([H_{2} CO_{3original}] = K [\mathrm{CO}_{2original}]\) Now, we can use the original concentration of carbonic acid to calculate the pH of the unpolluted rain 40 years ago: \(pH_{original} = -\log_{10} [H^{+}_{original}]\)

Step 4: (b) Calculate the dissolved CO₂ volume in today's rainwater.

We have the CO₂ concentration in today's rainwater. We can use the ideal gas law to find the volume of CO₂ at 25°C and 101.3 kPa in a 20.0-L bucket of today's rainwater. The ideal gas law is given by: \(PV = nRT\) where P is the pressure, V is the volume, R is the ideal gas constant, T is the temperature, and n is the number of moles. We can rearrange this equation to find the volume (V): \(V = \frac{nRT}{P}\) Now, we can use the given temperature and pressure, along with the concentration of CO₂, to find the volume of CO₂ dissolved in the rainwater: \(V_{CO_2} = \frac{[CO_2] \cdot 20.0 \mathrm{L} \cdot 8.314 \mathrm{J/mol\cdot K} \cdot 298 \mathrm{K}}{101.3 \mathrm{kPa}}\) Calculate the volume of CO₂ in today's rainwater using the above formula.

Key Concepts

Carbonic acid formation

In the atmosphere, carbon dioxide (\(\mathrm{CO}_2\)) has a significant impact on rainwater's acidity through a process called carbonic acid formation. When \(\mathrm{CO}_2\) from the air dissolves in water, it reacts to form carbonic acid \(\mathrm{H}_2\mathrm{CO}_3\). This process can be represented by the chemical equation:\[\mathrm{CO}_{2}(g)+\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{H}_{2} \mathrm{CO}_{3}(a q)\]Several key points to understand here include:

• Carbonic acid is a weak acid that partially dissociates into hydrogen ions \((\mathrm{H}^+)\) and bicarbonate ions \((\mathrm{HCO}_3^-)\).
• The presence of \(\mathrm{H}^+\) ions in the solution is what makes the water acidic, influencing its pH level.
• The higher the \(\mathrm{CO}_2\) concentration in the atmosphere, the more acidic the rain tends to be because more carbonic acid forms.

The formation of carbonic acid is a natural and essential process impacting environmental pH levels, and understanding it is crucial in studying the biogeochemical cycles on Earth.

Rainwater pH

pH is a scale that measures the acidity or basicity of a solution, ranging from 0 to 14. A pH of 7 is neutral, below 7 is acidic, and above 7 is basic.
For rainwater, the pH can vary depending on environmental factors, primarily the presence of acidic substances.
Today's Unpolluted Rainwater
The average pH of clean, unpolluted rain today is around 5.4. This is slightly acidic for a few reasons:

• The dissolved \(\mathrm{CO}_2\) in the rainwater forms carbonic acid, making the solution slightly acidic.
• Even clean rainwater naturally has some level of acidity due to atmospheric \(\mathrm{CO}_2\).

Rainwater 40 Years Ago
Forty years ago, the atmospheric \(\mathrm{CO}_2\) levels were lower, at 320 ppm compared to today's 400 ppm. This reduction implies that the pH of rainwater would have been higher, indicating less acidity.
Calculations based on this information allow us to backtrack and estimate the pH of rainwater from the past by using the established relation between \([\mathrm{H}_2\mathrm{CO}_3]\) and \([\mathrm{H}^+]\). Understanding how atmospheric conditions influence rainwater pH is vital for studies on climate change and environmental science.

Ideal gas law

The ideal gas law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and number of moles of a gas.The equation is written as:\[PV = nRT\]Where:

• \(P\) is the pressure
• \(V\) is the volume
• \(n\) is the number of moles
• \(R\) is the ideal gas constant (8.314 J/mol·K)
• \(T\) is the temperature in Kelvin

In the context of the exercise, we use the ideal gas law to determine how much \(\mathrm{CO}_2\) is dissolved in today's rainwater.
For example, by knowing the concentration of \(\mathrm{CO}_2\) and given conditions like temperature (25°C converted to Kelvin) and pressure (101.3 kPa), we can calculate the volume of \(\mathrm{CO}_2\) in a specific volume of rainwater, such as a 20.0-L bucket. This calculation illustrates how gases behave under certain conditions and is often applied in environmental and physical sciences to study gas solubility and its effects.