Problem 119 - Chemistry: The Central Science

Question

Atmospheric \(\mathrm{CO}_{2}\) levels have risen by nearly \(20 \%\) over the past 40 years from \(315 \mathrm{ppm}\) to \(380 \mathrm{ppm}\). (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 \(1.0 \mathrm{~atm}\) is dissolved in a 20.0-L bucket of today's rainwater?

Step-by-Step Solution

Answer

(a) The pH of unpolluted rain 40 years ago is approximately -log(\(304 \times \frac{10^{(-5.4)}}{380}\)). (b) The volume of CO₂ in a 20.0-L bucket of today's rainwater is \(\frac{(3.8 \times 10^{-4} mol/L \times 20L)(0.0821 L atm/mol K)(298 K)}{1 atm}\) L.

Step 1: (a) Find the pH of unpolluted rain 40 years ago

1. Determine the CO2 concentration 40 years ago: Since CO2 levels have risen by 20%, CO2 concentration 40 years ago = Current concentration - 0.20 × Current concentration Current concentration = \(380 ppm\) \(CO2_{40years\_ago} = 380 - 0.20 \times 380 = 304 ppm\) 2. Determine the H₂CO₃ concentration: The reaction between CO2 and H2O is in equilibrium. As the CO2 concentration increases, the H₂CO₃ concentration increases as well. We assume that the ratio of CO2 concentration to H₂CO₃ concentration is constant. \(\frac{H_{2}CO_{3_{today}}}{CO_{2_{today}}} = \frac{H_{2}CO_{3_{40years\_ago}}}{CO_{2_{40years\_ago}}}\) Given the pH of today's rainwater, we can calculate the concentration of H₂CO₃: pH = -log(\(H_{3}O^{+}\)), Thus, \(H_{3}O^{+}\) = \(10^{(-5.4)}\) The dissociation of carbonic acid H₂CO₃ forms H₃O⁺ and HCO₃⁻ ions. Therefore, \(H_{2}CO_{3_{today}} = H_{3}O^{+}_{today}\). Now we can find the concentration of H₂CO₃ in rainwater 40 years ago. \(H_{2}CO_{3_{40years\_ago}} = CO_{2_{40years\_ago}} \times \frac{H_{2}CO_{3_{today}}}{CO_{2_{today}}}\) \(H_{2}CO_{3_{40years\_ago}} = 304 \times \frac{10^{(-5.4)}}{380}\) 3. Calculate the pH of rain 40 years ago: The concentration of H₃O⁺ ions in the rainwater 40 years ago will be equal to the concentration of H₂CO₃ as it dissociates into H₃O⁺ ions. Therefore, \(H_{3}O^{+}_{40years\_ago} = H_{2}CO_{3_{40years\_ago}}\) pH = -log(\(H_{3}O^{+}_{40years\_ago}\)) pH = -log(\(304 \times \frac{10^{(-5.4)}}{380}\))

Step 2: (b) Calculate the volume of CO₂ in a 20.0-L bucket of today's rainwater

1. Calculate the moles of CO₂: First, convert the CO2 concentration from ppm to moles: \(CO_{2}\) concentration (mol/L) = \(CO_{2}\) concentration (ppm) × \(\frac{1mol}{10^6 L}\) \(CO_{2}\) concentration (mol/L) = \(380 ppm \times \frac{1 mol}{10^6 L} = 3.8 \times 10^{-4} mol/L\) Now calculate the moles of CO₂ in 20.0-L bucket of rainwater: Moles of CO₂ = \(CO_{2}\) concentration (mol/L) × Volume of rainwater (L) Moles of CO₂ = \(3.8 \times 10^{-4} mol/L \times 20L\) 2. Use the Ideal Gas Law to find the volume of CO₂: The Ideal Gas Law is: PV = nRT Where P = pressure (1 atm), V = volume (L), n = moles of CO₂, R = gas constant (0.0821 L atm/mol K), and T = temperature (298 K) Rearrange the formula and solve for volume V: V = \(\frac{nRT}{P}\) V = \(\frac{(3.8 \times 10^{-4} mol/L \times 20L)(0.0821 L atm/mol K)(298 K)}{1 atm}\)

Key Concepts

Atmospheric CO2 levels

Over the past 40 years, the levels of carbon dioxide (CO₂) in the atmosphere have increased significantly. This rise can have various implications on environmental and biological systems. Increased CO₂ levels, climbing from 315 parts per million (ppm) to 380 ppm, means there's more CO₂ available in the air to dissolve into water and form carbonic acid (H₂CO₃).
\[\text{CO}_2(g) + \text{H}_2\text{O}(l) \rightleftharpoons \text{H}_2\text{CO}_3(aq)\]
Why is this important? CO₂ plays a central role in the Earth's carbon cycle and has direct effects on climate change by trapping heat in the atmosphere. Higher CO₂ can also lead to acidification of oceans and freshwater systems due to increased formation of carbonic acid, which can lower pH levels. Understanding CO₂ levels is vital to assessing its impact on both natural systems and human health.

Rainwater pH

The pH of rainwater is an indication of its acidity or alkalinity. It's primarily influenced by the presence of CO₂ in the atmosphere. When CO₂ dissolves in rainwater, it forms carbonic acid, contributing to a lower pH level.
In unpolluted conditions, rainwater typically has a pH around 5.6 due to naturally occurring carbonic acid. Today's average rainwater pH of 5.4 suggests mild acidity, but 40 years ago, with lower CO₂ levels at 304 ppm, the rain would have been less acidic, meaning a higher pH.
To calculate the pH change over the years, we assume:

• Carbonic acid formation is the sole factor influencing pH.
• The proportion between CO₂ and H₂CO₃ remains constant.

This can be calculated using: \[\text{pH} = -\log([\text{H}_3\text{O}^+])\] Where \[\text{H}_3\text{O}^+\] represents the concentration of hydronium ions, directly linked to H₂CO₃ concentration. Greater atmospheric CO₂ leads to more carbonic acid, shifting the equilibrium towards increased hydronium ions and thereby lowering the pH.

Ideal Gas Law

The Ideal Gas Law is a critical equation in chemistry, which relates the state of a gas to its pressure, volume, temperature, and amount. It's expressed as:\[PV = nRT\]Where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles of gas, \(R\) is the ideal gas constant (0.0821 L atm/mol K), and \(T\) is the temperature in Kelvin.

This equation helps to determine the volume of CO₂ dissolved in rainwater at specific conditions. Suppose we have rainwater containing 3.8 x 10⁻⁴ mol/L of CO₂ in a 20.0-L bucket at conditions of 25°C and 1 atm pressure.

• First, we calculate moles of CO₂ by multiplying the concentration by the volume of the rainwater.
• Next, rearrange the Ideal Gas Law to solve for volume:
• \[V = \frac{nRT}{P}\]

The calculated volume is a measure of how much space this dissolved gas would occupy if it were in a purely gaseous state, offering insight into how gases behave under various conditions.