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 \(\mathrm{CO}_{2}(\mathrm{~g})\) 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). (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

(a) The chemical equation for the neutralization reaction between sodium bicarbonate and lactic acid is: NaHCO3 + CH3CH(OH)COOH -> CH3CH(OH)COONa + H2O + CO2. (b) The concentration of lactic acid in one cup of sour milk is approximately 0.54 mol/L. (c) The volume of carbon dioxide gas produced at 1 atm pressure and 350°F is approximately 4.59 L.

Step-by-step solution

(a) Write the chemical equation for the neutralization reaction

To write the chemical equation, we need to know the reactants and products. We are given the reactants as sodium bicarbonate (NaHCO3) and lactic acid (CH3CH(OH)COOH). The neutralization reaction between an acid and a base (in this case, the sodium bicarbonate) usually produces a salt, water, and carbon dioxide gas. In this specific reaction, the products will be sodium lactate (formed from the sodium ion and the lactate ion), water, and carbon dioxide gas: NaHCO3 + CH3CH(OH)COOH -> CH3CH(OH)COONa + H2O + CO2.

(b) Calculate the concentration of lactic acid in one cup of sour milk

We're given that 1/2 teaspoon of baking soda is neutralized by one cup of sour milk. As the density of baking soda is 2.16 g/cm^3, we can find the amount of baking soda in one half-teaspoon. Since there are 48 teaspoons in 236.6 mL, there are 4.925 mL in half a teaspoon. Therefore, the mass of baking soda is: \(4.925 \mathrm{~mL} \times \frac{2.16 \mathrm{~g}}{\mathrm{1 cm^3}} \times \frac{\mathrm{1 cm^3}}{\mathrm{1 mL}}= 10.64\mathrm{~g}\) Now, we can convert this mass into moles using the molar mass of NaHCO3 (84 g/mol): \(\frac{10.64 \mathrm{~g}}{84 \mathrm{~g/mol}} = 0.1267\mathrm{~mol}\) of baking soda Since the neutralization reaction has a 1:1 ratio between baking soda and lactic acid, there are also 0.1267 mol of lactic acid in one cup of sour milk. To get the concentration in mol/L, we divide the moles of lactic acid by the volume of one cup of sour milk in liters: \(\frac{0.1267\mathrm{~mol}}{0.2366\mathrm{~L}} = 0.5357 \mathrm{~mol/L}\) So, the concentration of lactic acid in one cup of sour milk is about 0.54 mol/L.

(c) Calculate the volume of carbon dioxide gas produced

In the neutralization reaction, we have a 1:1 ratio between baking soda and carbon dioxide gas. Since there are 0.1267 mol of baking soda, there will be 0.1267 mol of CO2 produced. We can use the ideal gas law to calculate the volume of CO2 produced. The ideal gas law is: PV = nRT We are given the temperature (350°F) and pressure (1 atm). First, we need to convert the temperature to Kelvin: \(350^\circ F = 176.7^\circ C = 449.8 K\) Now, we can plug the values into the ideal gas law: \(V = \frac{nRT}{P} = \frac{0.1267\mathrm{~mol} \times 0.08206\mathrm{~L\, atm/mol\, K} \times 449.8\mathrm{~K}}{1\mathrm{~atm}}\) \(V = 4.591\mathrm{~L}\) The volume of carbon dioxide gas produced at 1 atm pressure and 350°F is approximately 4.59 L.

Key concepts

Chemical Equation

Chemical equations are expressions that showcase chemical reactions where the reactants transform into products. In this context, we are looking at the neutralization reaction between baking soda (sodium bicarbonate, \(\mathrm{NaHCO}_3\)) and lactic acid \(\mathrm{CH}_3\mathrm{CH(OH)COOH}\). During a neutralization reaction, an acid reacts with a base to produce a salt, water, and sometimes a gas. Here, when sodium bicarbonate and lactic acid react, the products are sodium lactate, water, and carbon dioxide gas.
The balanced chemical equation is:
\[\mathrm{NaHCO}_3 + \mathrm{CH}_3\mathrm{CH(OH)COOH} \rightarrow \mathrm{CH}_3\mathrm{CH(OH)COONa} + \mathrm{H}_2\mathrm{O} + \mathrm{CO}_2\]
The importance of balancing this equation lies in confirming that the reactants and products are in accordance with the law of conservation of mass. That means no atoms are lost or gained, they are merely rearranged. This is a core principle of chemistry and essential for predicting the amounts of products formed.

Lactic Acid

Lactic acid is a naturally occurring carboxylic acid with the formula \(\mathrm{CH}_3\mathrm{CH(OH)COOH}\). It is commonly found in sour milk, yogurt, and other fermented dairy products. In this exercise, lactic acid is the acid component reacting with baking soda to cause a chemical change.
The presence of lactic acid in sour milk is why baking soda works so effectively as a leavening agent when making baked goods. When lactic acid comes in contact with baking soda, it causes a reaction that releases carbon dioxide gas. This gas forms bubbles in the dough or batter, causing it to expand and rise.

• Lactic acid is not only an important component in baking but is also crucial in various fermentation processes.
• Its solution in sour milk acts as a natural preservative and flavoring agent, contributing to the tangy taste.

Understanding the role of lactic acid in reactions with bases like sodium bicarbonate underpins its use in baking and food science.

Ideal Gas Law

The ideal gas law is a vital equation in chemistry used to relate the pressure, volume, temperature, and amount of gas in a system. The equation is expressed as \(PV = nRT\). Here,

• \(P\) represents the pressure of the gas. This is measured in atmospheres (atm).
• \(V\) is the volume occupied by the gas in liters (L).
• \(n\) stands for the number of moles of the gas.
• \(R\) is the ideal gas constant, approximately 0.08206 L atm/mol K.
• \(T\) signifies the temperature which needs to be in Kelvin (K).

When solving for the volume of carbon dioxide gas produced from the baking soda and lactic acid reaction, knowing these variables allows you to predict the gas volume under specific conditions.
First, make sure to convert the temperature from Fahrenheit to Kelvin to ensure accurate calculations:
\[350^\circ \mathrm{F} = 176.7^\circ \mathrm{C} = 449.8 \,\mathrm{K}\]
Substituting the known values into the ideal gas law gives an accurate measure of the carbon dioxide gas generated. This illustrates the direct and practical uses of the ideal gas law in predicting outcomes from baking and other real-life applications.