A baker is making strawberry cupcakes and wants to ensure they are very light. To do this, he must add enough baking soda to increase the volume of the cupcakes by \(55.0 \%\). All of the carbon dioxide produced by the decomposition of the baking soda is incorporated into the cupcakes. The baking soda or sodium bicarbonate will react with citric acid in the batter according to the reaction \(3 \mathrm{NaHCO}_{3}+\mathrm{H}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7} \longrightarrow \mathrm{Na}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}+3 \mathrm{H}_{2} \mathrm{O}+3 \mathrm{CO}_{2}\) which occurs with a \(63.7 \%\) yield. How many grams of baking soda must be added to \(1.32 \mathrm{~L}\) of cupcake batter at a baking temperature of \(325^{\circ} \mathrm{F}\) and a pressure of 738 torr to achieve the correct consistency?

The Ideal Gas Law is a fundamental concept in chemistry that relates the pressure, volume, temperature, and number of moles of a gas. The formula is expressed as \( PV = nRT \), where:

• \( P \) is the pressure of the gas.
• \( V \) is the volume of the gas.
• \( n \) is the number of moles of the gas.
• \( R \) is the universal gas constant \( (0.0821 \text{ L} \, \text{atm} / (\text{mol} \, \text{K})) \).
• \( T \) is the temperature in Kelvin.

To use the Ideal Gas Law, you must ensure all units are in the correct form: pressure in atmospheres, volume in liters, and temperature in Kelvin. In this problem, we used the Ideal Gas Law to determine the moles of carbon dioxide \( (\text{CO}_2) \) by rearranging the formula to \( n = \frac{PV}{RT} \). This helped us understand how much \( \text{CO}_2 \) is produced, which is essential for finding out how much baking soda is needed for the cupcake batter.

Stoichiometry is the calculation of reactants and products in chemical reactions. It allows us to predict how much product we can produce from a given amount of reactant. In this exercise, we used stoichiometry to relate the amount of sodium bicarbonate \( (\text{NaHCO}_3) \) to the carbon dioxide \( (\text{CO}_2) \) produced. The balanced chemical equation for the reaction:\[ 3 \,\text{NaHCO}_3 + \text{H}_3 \text{C}_6 \text{H}_5 \text{O}_7 \rightarrow \text{Na}_3 \text{C}_6 \text{H}_5 \text{O}_7 + 3 \, \text{H}_2 \text{O} + 3 \, \text{CO}_2 \]makes it clear that 3 moles of \text{NaHCO}_3 produce 3 moles of \text{CO}_2. This 1:1 molar ratio simplifies our calculations. We calculated the moles of \text{CO}_2 using the Ideal Gas Law, and then used stoichiometry to find how many moles of \text{NaHCO}_3 are required to produce that amount of \text{CO}_2.

Understanding chemical reactions is crucial for solving problems involving chemical changes. A chemical reaction involves the rearrangement of atoms to transform reactants into products. The balanced chemical equation provided:\[ 3 \, \text{NaHCO}_3 + \text{H}_3 \text{C}_6 \text{H}_5 \text{O}_7 \rightarrow \text{Na}_3 \text{C}_6 \text{H}_5 \text{O}_7 + 3 \, \text{H}_2 \text{O} + 3 \, \text{CO}_2 \]describes the reaction between sodium bicarbonate and citric acid. In this equation, reactants \text{NaHCO}_3 and \text{H}_3 \text{C}_6 \text{H}_5 \text{O}_7dissociate to form products including \text{CO}_2, which helps increase the cupcake volume. Understanding the stoichiometric coefficients (numbers in front of compounds) helps us balance the reaction and apply stoichiometric principles to determine quantities of reactants and products.

Unit conversion is vital in chemistry to ensure consistency and accuracy in calculations. The problem required converting temperature and pressure into standard units. The temperature was converted from Fahrenheit to Kelvin:
\( T(K) = (325 - 32) \times \frac{5}{9} + 273.15 = 436.85 \text{ K} \).
The pressure needed to be in atmospheres (atm) instead of torr:
\( P = 738 \, \text{torr} \times \frac{1 \, \text{atm}}{760 \, \text{torr}} = 0.971 \, \text{atm} \. \).
These conversions placed all variables in the Ideal Gas Law formula (\(PV = nRT\)) in the correct units, allowing for accurate calculation of \text{CO}_2 moles. Always check units before using formulas.

Yield calculation tells us the efficiency of a chemical reaction. The 'percent yield' indicates the ratio of actual yield to theoretical yield, multiplied by 100. In the given reaction, the yield was 63.7%, meaning only 63.7% of the possible reaction occurred. To find out how much \text{NaHCO}_3 to use, we adjusted for yield:
\( \text{Moles of NaHCO}_3 = \frac{0.0193 \, \text{moles CO}_2}{0.637} \approx 0.0303 \, \text{moles NaHCO}_3 \. \).
Finally, we converted moles to grams using the molar mass of \text{NaHCO}_3 (84 g/mol):
\( \text{Mass} = 0.0303 \, \text{moles} \times 84 \, \text{g/mol} = 2.5 \, \text{g} \. \).
Understanding yield allows us to account for less-than-perfect reaction conditions and still achieve the desired outcome.