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. \((\mathbf{a})\) 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 \() .(\mathbf{c})\) 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

01

Write the balanced chemical equation for the neutralization reaction

Sodium bicarbonate reacts with lactic acid to form water, carbon dioxide, and sodium lactate. The unbalanced chemical equation for this reaction is: NaHCO3 + CH3CH(OH)COOH -> H2O + CO2 + CH3CH(OH)COONa Since there's an equal amount of atoms for each element on both sides, the equation is already balanced.

02

Calculate the concentration of lactic acid in one cup of sour milk

Firstly, convert the density of baking soda to mass. According to the given information, 1/2 teaspoon of baking soda is enough to neutralize one cup of sour milk. The density of baking soda is 2.16 g/cm³. Volume of 1/2 teaspoon of baking soda = 1/2 * 4.93 cm³ = 2.465 cm³ Mass of 1/2 teaspoon of baking soda = Volume * Density = 2.465 cm³ * 2.16 g/cm³ = 5.3246 g Now, convert the mass of baking soda to moles: Molar mass of NaHCO3 = 23(g/mol) + 1(g/mol) + 12(g/mol) + 3 * 16(g/mol) = 84 g/mol Moles of NaHCO3 = 5.3246 g / 84 g/mol ≈ 0.06339 moles Since 1 mole of NaHCO3 neutralizes 1 mole of lactic acid: moles of lactic acid = moles of NaHCO3 = 0.06339 moles Now, calculate the concentration of lactic acid in one cup of sour milk (=236.6 mL): Concentration of lactic acid = moles / volume = 0.06339 moles / 236.6 mL = 0.2679 mol/L

03

Calculate the volume of carbon dioxide gas produced

To find the volume of carbon dioxide gas produced, we will use the ideal gas equation: PV = nRT Where: P = pressure = 101.3 kPa n = moles of CO2 formed = 0.06339 moles (since 1 mole of NaHCO3 produces 1 mole of CO2) R = gas constant = 8.314 J/(mol·K) (converting kPa to J, multiply by 1000: 8.314 * 1000 = 8.314 kPa·L/(mol·K)) T = temperature = 177°C + 273.15 = 450.15 K Rearrange the formula to solve for the volume, V: V = nRT/P V = (0.06339 moles) × (8.314 kPa·L/(mol·K)) × (450.15 K) / (101.3 kPa) V ≈ 0.1052 L The volume of carbon dioxide gas produced is approximately 0.1052 L, at a pressure of 101.3 kPa and a temperature of 177°C.

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