Carbon dioxide can be made in the lab by the reaction of hydrochloric acid with calcium carbonate. How many milliliters of \(\mathrm{dry} \mathrm{CO}_{2}\) at \(20.0^{\circ} \mathrm{C}\) and 745 torr can be prepared from a mixture of \(12.3 \mathrm{~g}\) of \(\mathrm{CaCO}_{3}\) and \(185 \mathrm{~mL}\) of \(0.250 \mathrm{MHCl} ?\)

Understanding the process of balancing chemical equations is crucial for studying gas-forming reactions. It involves ensuring that the number of atoms for each element is the same on both sides of the reaction. For instance, the reaction between hydrochloric acid (HCl) and calcium carbonate (CaCO3) to produce carbon dioxide (CO2) is expressed as:

\[ \text{CaCO}_3 (\text{s}) + 2\text{HCl} (\text{aq}) \rightarrow \text{CO}_2 (\text{g}) + \text{H}_2\text{O} (\text{l}) + \text{CaCl}_2 (\text{aq}) \]
Notice that the coefficients of the reactants and products are selected to ensure the conservation of mass. This means that the number of calcium, carbon, oxygen, hydrogen, and chlorine atoms are balanced. The chemical equation provides a ratio that's essential for stoichiometric calculations, as it allows us to relate moles of reactants to moles of products.

Moles are a fundamental concept in chemistry, representing the quantity of a substance. One mole corresponds to Avogadro's number (\(6.022 \times 10^{23}\) entities) of atoms or molecules. To calculate moles in chemical reactions, you will often use the molar mass of a substance – the mass of one mole of that substance. For example,

\[ \text{moles of CaCO}_3 = \frac{12.3\text{ g}}{100.09\text{ g/mol}} \]
Likewise, the moles of a solution, like hydrochloric acid (HCl), are calculated with its molarity (M, or moles per liter) and volume (L).

\[ \text{moles of HCl} = 0.250\text{ M} \times 0.185\text{ L} \]
Knowing the moles of each reactant allows us to use stoichiometry to relate reactants to products.

The limiting reactant in a chemical reaction is the substance that will be used up first, thus determining the amount of products formed. Identifying the limiting reactant is akin to finding out which ingredient will run out first when cooking a meal. In our example reaction, once the moles of both reactants have been calculated, we compare them to the mole ratio from the balanced chemical equation. With a 2:1 ratio of HCl to CaCO3,

\[ \frac{\text{moles of HCl}}{\text{moles of CaCO}_3} > 2 \]
indicates HCl is in excess and CaCO3 is the limiting reactant. This is powerful information since the moles of all products formed can be predicted using just the number of moles of the limiting reactant.

The ideal gas law relates the volume (V), pressure (P), number of moles (n), and temperature (T) of a gas. Expressed as \( PV = nRT \) where R is the ideal gas constant (\( 0.0821\text{ L} \cdot \text{atm} / (\text{mol} \cdot \text{K}) \) ), the ideal gas law allows us to determine the volume of a gas under specific conditions. To solve for the volume of CO2 produced in our reaction at 20.0°C and 745 torr, we first convert the temperature to Kelvin and the pressure to atmospheres. Then, we can rearrange the law to solve for V:

\[ V = \frac{nRT}{P} \]
Here, inserting the calculated moles of CO2, the given temperature in Kelvin and pressure in atmospheres, the volume of dry CO2 can be determined. This step is the final piece in connecting the mass of a solid reactant with the volume of a gas product, a frequent challenge in stoichiometric gas-forming reactions.