Problem 36 - Chemistry and Chemical Reactivity

Question

5\. A self-contained underwater breathing apparatus (SCUBA) uses canisters containing potassium superoxide. The superoxide consumes the \(\mathrm{CO}_{2}\) exhaled by a person and replaces it with oxygen. $$ 4 \mathrm{KO}_{2}(\mathrm{s})+2 \mathrm{CO}_{2}(\mathrm{g}) \rightarrow 2 \mathrm{K}_{2} \mathrm{CO}_{3}(\mathrm{s})+3 \mathrm{O}_{2}(\mathrm{g}) $$ What mass of \(\mathrm{KO}_{2}\), in grams, is required to react with \(8.90 \mathrm{L}\) of \(\mathrm{CO}_{2}\) at \(22.0^{\circ} \mathrm{C}\) and \(767 \mathrm{mm} \mathrm{Hg} ?\)

Step-by-Step Solution

Answer

52.20 grams of \(\mathrm{KO}_2\) is required.

Step 1: Use Ideal Gas Law

We need to find the number of moles of \(\mathrm{CO}_2\) using the ideal gas law equation, \(PV = nRT\). We know \(P = 767 \ \mathrm{mmHg} = 767/760 \ \mathrm{atm}\), \(V = 8.90 \ \mathrm{L}\), \(R = 0.0821 \ \mathrm{L} \cdot \mathrm{atm}/\mathrm{mol} \cdot \mathrm{K}\), and \(T = 22.0^{\circ} \mathrm{C} = 295 \ \mathrm{K}\). Substitute these values into the equation and solve for \(n\).

Step 2: Calculate Moles of CO2

Substitute the values into the ideal gas law: \[ n = \frac{PV}{RT} = \frac{(767/760) \times 8.90}{0.0821 \times 295} \approx 0.367 \ \mathrm{mol \ \mathrm{CO}_2} \]

Step 3: Use Stoichiometry to Find Moles of KO2

From the balanced equation, \(4 \ \mathrm{KO}_2\) reacts with \(2 \ \mathrm{CO}_2\). Thus, \(\frac{4}{2} = 2\) moles of \(\mathrm{KO}_2\) are needed per mole of \(\mathrm{CO}_2\). Multiply the moles of \(\mathrm{CO}_2\) by 2 to find moles of \(\mathrm{KO}_2\):\[ 0.367 \times 2 = 0.734 \ \mathrm{mol \ \mathrm{KO}_2} \]

Step 4: Calculate Mass of KO2

Find the molar mass of \(\mathrm{KO}_2\): - \(\mathrm{K}\) = 39.10 \ \mathrm{g/mol},- \(\mathrm{O}_2\) = 2 \times 16.00 = 32.00 \ \mathrm{g/mol}.Thus, the molar mass of \(\mathrm{KO}_2\) is \(39.10 + 32.00 = 71.10 \ \mathrm{g/mol}\). Use this to calculate the mass of \(\mathrm{KO}_2\):\[ 0.734 \ \mathrm{mol} \times 71.10 \ \mathrm{g/mol} = 52.20 \ \mathrm{g} \]

Step 5: Conclude the Mass Required

Thus, the mass of \(\mathrm{KO}_2\) required to react with \(8.90\ \mathrm{L}\) of \(\mathrm{CO}_2\) under the given conditions is approximately \(52.20 \ \mathrm{g}\).

Key Concepts

Ideal Gas Law

The Ideal Gas Law is crucial for solving problems involving gases, like in our SCUBA system exercise. It establishes a relationship between pressure, volume, temperature, and moles of a gas, using the equation \(PV = nRT\). Here, *P* is pressure, *V* is volume, *n* is the number of moles, *R* is the ideal gas constant, and *T* is temperature in Kelvin.

Converting given conditions to suitable units is essential:

• Pressure (P) from mmHg to atm: 767 mmHg is equivalent to \(\frac{767}{760}\) atm.
• Temperature (T) from °C to Kelvin: Add 273 to °C to get Kelvin. Thus, 22°C becomes 295 K.
• Volume (V) is typically given in liters.

This law helps us calculate the moles of \(\mathrm{CO}_2\) by substituting these values into the equation, making features of gas behavior predictable and manageable.

Chemical Reaction

A chemical reaction describes a process where reactants are transformed into products, following the laws of conservation of mass and energy. In our exercise, the reaction with potassium superoxide \(\mathrm{KO}_2\) and carbon dioxide \(\mathrm{CO}_2\) is:

• The balanced equation is: \(4 \mathrm{KO}_2(s) + 2 \mathrm{CO}_2(g) \to 2 \mathrm{K}_2\mathrm{CO}_3(s) + 3 \mathrm{O}_2(g)\).
• It shows how many moles of each substance react and are formed.
• Stoichiometry is used to determine the proportions needed for complete reactions.

In this context, the reaction consumes \(\mathrm{CO}_2\) and produces \(\mathrm{O}_2\), making it useful in SCUBA systems to filter and renew breathable air.

It's essential to respect the coefficients in chemical equations, as they dictate how molecules interact.

Molar Mass

Molar mass is the mass of one mole of a substance, expressed in grams per mole \(\mathrm{g/mol}\). It acts as a bridge between the microscale (molecules) and macroscale (grams) in chemistry.

To find the molar mass of a compound like \(\mathrm{KO}_2\), we:

• Add the atomic masses of all atoms in the molecule:
  • \(\mathrm{K}\) (potassium) has an atomic mass of 39.10 g/mol.
  • \(\mathrm{O}_2\) (oxygen) has 2 atoms, each 16.00 g/mol, thus totaling 32.00 g/mol.
• Combine these to get \(39.10 + 32.00 = 71.10\) g/mol for \(\mathrm{KO}_2\).

Knowing the molar mass allows us to convert moles of \(\mathrm{KO}_2\) into grams, making calculation possible and practical for actual use in chemical reactions.

SCUBA Systems

SCUBA systems stand for Self-Contained Underwater Breathing Apparatus. These ingenious devices allow divers to breathe underwater by using canisters filled with substances like potassium superoxide \(\mathrm{KO}_2\).

Here's how they work:

• They absorb carbon dioxide \(\mathrm{CO}_2\) exhaled by the diver.
• The chemical reaction as illustrated with \(\mathrm{KO}_2\) takes \(\mathrm{CO}_2\) and releases \(\mathrm{O}_2\), allowing the diver to breathe safely.
• This system eliminates the need for conventional air tanks by enhancing air supply efficiency.

The balance of chemical reactions and law of stoichiometry ensure these systems are lightweight yet effective, making underwater exploration more accessible and safe.