Question

Baking soda (sodium bicarbonate, \(\mathrm{NaHCO}_{3}\)) reacts with acids in foods to form carbonic acid (\(\mathrm{H}_{2} \mathrm{CO}_{3}\)), 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} .\( Calculate the concentration of lactic acid in one cup of sour milk(assuming the rule of thumb applies), in units of mol/L. (One cup \)=236.6 \mathrm{mL}=48\( teaspoons). (c) If 1/2 teaspoon of baking soda is indeed completely neutralized by the lactic acid in sour milk, calculate the volume of carbon dioxide gas that would be produced at 1 atm pressure, in an oven set to \)350^{\circ} \mathrm{F}$ .

Short answer

The chemical equation for the neutralization reaction between sodium bicarbonate and lactic acid is: \(NaHCO_3 + CH_3CH(OH)COOH → NaCH_3CH(OH)COO + H_2O + CO_2\). The concentration of lactic acid in one cup of sour milk is approximately \(0.268\, mol/L\). The volume of carbon dioxide gas produced at 1 atm and \(350^{\circ} F\) is approximately \(2.307\, L\).

Step-by-step solution

Write the chemical equation for the neutralization reaction

To write the chemical equation for the neutralization reaction, we need to find the products of the reaction between sodium bicarbonate (NaHCO3) and lactic acid (CH3CH(OH)COOH). In a neutralization reaction, an acid reacts with a base to produce a salt and water. In this case, we have: NaHCO3 (sodium bicarbonate, acting as a base) + CH3CH(OH)COOH (lactic acid) → NaCH3CH(OH)COO (sodium lactate, the salt) + H2CO3 (carbonic acid) Now, carbonic acid (H2CO3) decomposes into water (H2O) and carbon dioxide (CO2): H2CO3 → H2O + CO2 The overall reaction can be written as: NaHCO3 + CH3CH(OH)COOH → NaCH3CH(OH)COO + H2O + CO2

Calculate the concentration of lactic acid in sour milk

Given that 1/2 teaspoon of baking soda is neutralized by one cup of sour milk, and the density of baking soda is 2.16 g/cm³, we can calculate the mass of baking soda in 1/2 teaspoon: 1/2 teaspoon = 1/2 * 4.93 cm³ = 2.465 cm³ (since 1 teaspoon = 4.93 cm³) Mass of baking soda = density * volume = 2.16 g/cm³ * 2.465 cm³ ≈ 5.324 g Now, we should find the moles of baking soda: Moles of NaHCO3 = (mass of baking soda) / (molar mass of NaHCO3) = 5.324 g / (23 + 1 + 12 + 16*3) g/mol = 5.324 g / 84 g/mol ≈ 0.0634 mol Using the stoichiometry from the balanced chemical equation (1:1 mole ratio between NaHCO3 and lactic acid), the moles of lactic acid in one cup of sour milk are equal to the moles of baking soda: Moles of lactic acid = moles of NaHCO3 = 0.0634 mol To find the concentration of lactic acid in sour milk, we should divide the moles of lactic acid by the volume of one cup of sour milk: Concentration of lactic acid = (moles of lactic acid) / (volume of sour milk) = 0.0634 mol / 236.6 mL = 0.0634 mol / 0.2366 L ≈ 0.268 mol/L

Calculate the volume of carbon dioxide gas produced

To find the volume of CO2 produced, we can use the ideal gas law equation, PV = nRT, where P is the pressure, V is the volume, n is the amount in moles, R is the ideal gas constant, and T is the temperature. First, we need to convert the oven temperature from Fahrenheit to Kelvin: Temperature in Kelvin (K) = (Temperature in Fahrenheit - 32) * (5/9) + 273.15 = (350 - 32) * (5/9) + 273.15 ≈ 450.37 K Using the balanced chemical equation, there is a 1:1 mole ratio between NaHCO3 and CO2. Therefore, the moles of CO2 produced are equal to the moles of NaHCO3: Moles of CO2 = moles of NaHCO3 = 0.0634 mol Next, we can use the ideal gas law to calculate the volume of CO2: PV = nRT => V= nRT/P Assuming the pressure is 1 atm, the ideal gas constant (R) is 0.0821 L*atm/mol*K, and using the moles and temperature calculated earlier: V = 0.0634 mol * 0.0821 L*atm/mol*K * 450.37 K / 1 atm ≈ 2.307 L Therefore, the volume of carbon dioxide gas produced at 1 atm and 350 degrees Fahrenheit is approximately 2.307 L.

Key concepts

Understanding Stoichiometry

Stoichiometry is like a recipe for chemistry; it's all about the proportions of ingredients, which in this case are elements and compounds. In a baking scenario, just like we need a specific amount of flour and sugar to make a perfect cake, in a chemical reaction, we need exact amounts of reactants to fully react without leaving anything unreacted. For the neutralization reaction between sodium bicarbonate (baking soda) and lactic acid, stoichiometry tells us how much of each reactant is required to produce the desired products, water and carbon dioxide, without any excess.

As our calculation steps in the exercise demonstrated, we first calculated the moles of baking soda and then used the 1:1 mole ratio (from the balanced equation of the reaction) to determine the moles of lactic acid. This essential stoichiometric concept ensures that all the sodium bicarbonate is neutralized by lactic acid in our cake batter, leading to the perfect rise of the cake. Simplifying stoichiometry, it's about the 'mole-to-mole' comparison between reactants and products, which is crucial in predicting the outcome of our reaction and in this case, the success of our cake!

Decoding the Ideal Gas Law

The Ideal Gas Law is depicted by the simple yet elegant equation: PV = nRT. This relationship lets us juggle the variables of pressure (P), volume (V), temperature (T), and the number of moles (n) of a gas, along with R, the ideal gas constant, which is the same for all gases under ideal conditions. It's like knowing exactly the right temperature and time to bake a cake leads to a perfect fluffy texture; similarly, knowing the conditions of a gas helps predict how it will behave.

In our exercise, we used this law to find the volume of carbon dioxide at a specific temperature and pressure. After converting the temperature to Kelvin (since gas laws require an absolute temperature scale), we plugged in the values into the equation and found our answer. This calculation is pivotal, especially in scenarios where we need the actual volume of gas produced, like knowing how much our cake will rise with the produced CO2. In essence, this Ideal Gas Law is a powerful tool that relates the physical properties of an ideal gas, which, although an approximation, gives us invaluable insight into the behavior of real gases.

Making Sense of Molarity Calculations

Molarity, often denoted by 'M,' measures the concentration of a solution, telling us how much of a substance is dissolved in a certain volume of solvent. It's akin to understanding how sweet our cake will be, depending on how much sugar we've mixed into the batter. In chemistry, this is crucial because reactions happen when molecules collide, and molarity gives us an insight into how many molecules of reactants are present to potentially react.

Let's look back at our exercise. After finding the moles of lactic acid that would react with the given amount of baking soda, we divided these moles by the volume of the sour milk in liters to find the molarity. The result gives us a clear picture of the strength of our sour milk as an acid - essentially how 'sour' it is. Molarity calculations provide a clear, quantitative measure of the sour milk's acidity, guiding bakers to ensure the right rise of their cake by adjusting the amount of sour milk, or in a lab, allowing chemists to control the course of their reactions.