Pressure of CO₂ to Maintain 0.12 M at 25 °C Summary & Study Notes

🧪 Problem overview

You are asked to find the pressure of $CO_2$ required to keep the dissolved $CO_2$ concentration at $0.12\ \text{M}$ in a bottle of club soda at $25\ ^\circ\text{C}$. This uses Henry's law, which relates gas solubility to partial pressure.

📐 Key concept: Henry's law

Henry's law states: $C = k_H,P$, where $C$ is the dissolved gas concentration (in M), $P$ is the partial pressure of the gas (in atm), and $k_H$ is the Henry's law constant for that gas at the given temperature (in $\text{M·atm}^{-1}$).

For $CO_2$ at $25\ ^\circ\text{C}$ a commonly used value is $k_H\approx 0.034\ \text{M·atm}^{-1}$ (note: $k_H$ is temperature dependent).

🧮 Calculation

Rearrange Henry's law to solve for pressure:

$P = \frac{C}{k_H}$

Substitute the values:

$P = \frac{0.12\ \text{M}}{0.034\ \text{M·atm}^{-1}} = 3.53\ \text{atm}$

Convert to psi (optional):

$3.53\ \text{atm} \times 14.696\ \text{psi·atm}^{-1} \approx 51.9\ \text{psi}$

Rounded, the required partial pressure is about $3.5\ \text{atm}$ (≈ $52\ \text{psi}$).

🔍 Assumptions and notes

• The calculation assumes ideal behavior and that Henry's law applies (low to moderate concentrations, and constant temperature).
• The value of $k_H$ varies slightly between sources; using $k_H = 0.0339\ \text{M·atm}^{-1}$ gives $P\approx 3.54\ \text{atm}$, so results are robust to small changes in $k_H$.
• Real club soda contains other solutes (salts, sugars, flavorings) that can slightly change gas solubility; therefore this is an approximate value for typical bottled carbonated water.

✅ Final answer

The required partial pressure of $CO_2$ to maintain $0.12\ \text{M}$ dissolved $CO_2$ at $25\ ^\circ\text{C}$ is approximately $3.53\ \text{atm}$ (about $52\ \text{psi}$), using $k_H \approx 0.034\ \text{M·atm}^{-1}$.