Problem 119 - Chemistry The Central Science In Si Units

Question

Baking soda (sodium bicarbonate, \(\mathrm{NaHCO}_{3}\) ) reacts with acids in foods to form carbonic acid \(\left(\mathrm{H}_{2} \mathrm{CO}_{3}\right),\) which in turn decomposes to water and carbon dioxide gas. In a cake batter, the \(\mathrm{CO}_{2}(g)\) forms bubbles and causes the cake to rise, (a) A rule of thumb in baking is that \(1 / 2\) teaspoon of baking soda is neutralized by one cup of sour milk. The acid component in sour milk is lactic acid, \(\mathrm{CH}_{3} \mathrm{CH}(\mathrm{OH}) \mathrm{COOH}\). Write the chemical equation for this neutralization reaction. (b) The density of baking soda is \(2.16 \mathrm{~g} / \mathrm{cm}^{3}\). Calculate the concentration of lactic acid in one cup of sour milk (assuming the rule of thumb applies), in units of \(\mathrm{mol} / \mathrm{L}\). (One cup \(=236.6 \mathrm{~mL}=48\) teaspoons). \((\mathbf{c})\) If \(1 / 2\) teaspoon of baking soda is indeed completely neutralized by the lactic acid in sour milk, calculate the volume of carbon dioxide gas that would be produced at a pressure of \(101.3 \mathrm{kPa}\), in an oven set to \(177^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

a) \( \mathrm{NaHCO}_{3}(s) + \mathrm{CH}_{3} \mathrm{CH}(\mathrm{OH}) \mathrm{COOH}(aq) \rightarrow \mathrm{NaCH}_{3} \mathrm{CH}(\mathrm{OH}) \mathrm{COO}(aq) + \mathrm{H}_{2} \mathrm{CO}_{3}(aq) \). b) Molarity of lactic acid is 0.268 mol/L. c) Volume of CO\(_2\) produced is 23.33 L.

Step 1: Write the Chemical Equation

The neutralization reaction between baking soda (sodium bicarbonate, \( \mathrm{NaHCO}_{3} \)) and lactic acid (\( \mathrm{CH}_{3} \mathrm{CH}(\mathrm{OH}) \mathrm{COOH} \)) can be expressed as follows: \[ \mathrm{NaHCO}_{3}(s) + \mathrm{CH}_{3} \mathrm{CH}(\mathrm{OH}) \mathrm{COOH}(aq) \rightarrow \mathrm{NaCH}_{3} \mathrm{CH}(\mathrm{OH})\mathrm{COO}(aq) + \mathrm{H}_{2} \mathrm{CO}_{3}(aq) \] \( \mathrm{H}_{2} \mathrm{CO}_{3} \) is unstable and decomposes to water \( \mathrm{H}_2\mathrm{O} \) and carbon dioxide \( \mathrm{CO}_2 \): \[ \mathrm{H}_{2} \mathrm{CO}_{3} \rightarrow \mathrm{H}_{2} \mathrm{O} + \mathrm{CO}_{2}(g) \] This reveals how CO\(_2\) is released, causing the cake to rise.

Step 2: Convert Baking Soda to Grams and Calculate Moles

Given that \( \frac{1}{2} \) teaspoon of baking soda is used, first convert this to milliliters using the fact that 1 teaspoon equals 4.92892 mL. \[ \frac{1}{2} \text{ teaspoon} = 2.46446 \text{ mL} \] Now, use the density of baking soda (2.16 \( \text{g/cm}^3 \)) to find the mass: \[ \text{Mass} = \text{Density} \times \text{Volume} = 2.16 \text{ g/mL} \times 2.46446 \text{ mL} = 5.327 \text{ g} \] Calculate moles of \( \mathrm{NaHCO}_{3} \) using its molar mass (84.01 g/mol): \[ \text{Moles of } \mathrm{NaHCO}_{3} = \frac{5.327 \text{ g}}{84.01 \text{ g/mol}} = 0.0634 \text{ moles} \]

Step 3: Calculate Molarity of Lactic Acid in Sour Milk

From the given rule of thumb, \( \frac{1}{2} \) teaspoon of baking soda is neutralized by 1 cup of sour milk, indicating that \( 0.0634 \) moles of \( \mathrm{NaHCO}_{3} \) are neutralized by \( 236.6 \text{ mL} \) of sour milk. Calculate the molarity of lactic acid: \[ \text{Molarity} = \frac{\text{Moles of lactic acid}}{\text{Volume in Liters}} = \frac{0.0634 \text{ moles}}{0.2366 \text{ L}} = 0.268 \text{ mol/L} \]

Step 4: Calculate Volume of CO2 Produced

Using the ideal gas law \( PV = nRT \), where \( P = 101.3 \text{ kPa} \), \( n = 0.0634 \text{ moles} \), \( R = 8.314 \text{ J/(K·mol)} \), and \( T = 450.15 \text{ K} \) (177°C converted to Kelvin), calculate the volume of \( \mathrm{CO}_2 \): \[ V = \frac{nRT}{P} = \frac{0.0634 \times 8.314 \times 450.15}{101.3} \] Calculating this gives: \[ V \approx 23.33 \text{ L} \] Thus, about 23.33 liters of \( \mathrm{CO}_2 \) gas is produced.

Key Concepts

Neutralization

Neutralization occurs when an acid and a base react to form water and a salt. This type of reaction is vital in many chemical processes and real-world applications, such as baking. In the given example, baking soda (sodium bicarbonate, \( \mathrm{NaHCO}_{3} \)) serves as the base, while lactic acid (\( \mathrm{CH}_{3} \mathrm{CH(OH)COOH} \)), found in sour milk, is the acid. Load a cup of sour milk with half a teaspoon of baking soda and you'll find a chemical battle unfolding. This reaction yields sodium lactate, carbonic acid (which breaks down into carbon dioxide and water), a foamy array of outcomes that cause your cake to rise. This CO\(_2\) is what fills the batter with air, giving your dessert a fluffy finish. Watching this reaction take place is chemical magic you can nibble on!

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry connecting the pressure, volume, temperature, and moles of gas. It's expressed as \( PV = nRT \), where \( P \) stands for pressure, \( V \) is volume, \( n \) represents the moles of gas, \( R \) is the ideal gas constant, and \( T \) is temperature in Kelvin. This law assumes gases consist of particles that move randomly and occupy negligible space. This simplifies calculations, making it a super handy tool in baking science, such as calculating the volume of gas liberated in a reaction. In our baking exercise, it helps us determine the volume of \( \mathrm{CO}_2 \) produced at a given temperature and pressure, ensuring that you correctly gauge the lift your batter will boast.

Molarity Calculation

Molarity is a measure of concentration, expressed in moles per liter (mol/L). It's indispensable for understanding how much solute is present in a certain volume of solution.

• Firstly, gather the amount in moles of the compound.
• Next, divide by the volume of the solution in liters.

Imagine this calculation as your kitchen prep: gather all the ingredients (moles) and ensure they're mixed in the right balance (liters). In your baking soda-lactic acid reaction, calculate the moles of baking soda used, then relate it to the volume of sour milk. This determines how much acid is necessary to completely react, ensuring there's no leftover taste, only that perfect crumb lift.

Baking Soda Reaction

Baking soda reactions are simple yet magical. Beyond rising cakes, they teach us a lot about chemical balance and reactions. When \( \mathrm{NaHCO}_{3} \) interacts with an acid like \( \mathrm{CH}_{3} \mathrm{CH(OH)COOH} \), we see baking's science side.

• This reaction forms carbonic acid initially.
• Carbonic acid then decomposes into water and carbon dioxide.

This CO\(_2\) production is the push behind "rising" in baked goods. Think of it as tiny gas-filled balloons expanding and lightening your batter post-bake. Imagine your batter as a playground: baking soda is the fun-loving base that teams up with an acid to bring frothy fun to the mix! Knowing these fundamentals aids in giving your culinary creations that textbook rise every time.