Problem 100 - Chemistry An Atoms First Approach

Question

When 1.00 L of \(2.00 M \mathrm{Na}_{2} \mathrm{SO}_{4}\) solution at \(30.0^{\circ} \mathrm{C}\) is added to \(2.00 \mathrm{L}\) of \(0.750 M \mathrm{Ba}\left(\mathrm{NO}_{3}\right)_{2}\) solution at \(30.0^{\circ} \mathrm{C}\) in a calorimeter, a white solid (BaSO\(_{4}\)) forms. The temperature of the mixture increases to \(42.0^{\circ} \mathrm{C}\). Assuming that the specific heat capacity of the solution is \(6.37 \mathrm{J} /^{\circ} \mathrm{C} \cdot \mathrm{g}\) and that the density of the final solution is \(2.00 \mathrm{g} / \mathrm{mL},\) calculate the enthalpy change per mole of BaSO\(_{4}\) formed.

Step-by-Step Solution

Answer

The enthalpy change per mole of BaSO4 formed is 305,776 J/mol.

Step 1: Calculate the moles of Na2SO4 and Ba(NO3)2 initially present

We can use the formula: moles = volume × concentration moles of Na2SO4 = \(1.00 L × 2.00 M = 2.00\) moles moles of Ba(NO3)2 = \(2.00 L × 0.750 M = 1.50\) moles

Step 2: Determine the limiting reactant and calculate the moles of BaSO4 formed

The balanced reaction is: \(\mathrm{Na}_{2}\mathrm{SO}_{4}(aq) + \mathrm{Ba}(\mathrm{NO}_{3})_{2}(aq) \rightarrow \mathrm{BaSO}_{4}(s) + 2\mathrm{NaNO}_{3}(aq)\) Comparing the ratio of moles of reactants to the stoichiometric ratio, we find that: \(2.00 / 1 = 2.00\) \(1.50 / 1 = 1.50\) Since 1.50 is the lowest value, Ba(NO3)2 is the limiting reactant. Now, we calculate the moles of BaSO4 formed based on the stoichiometric ratios: moles of BaSO4 = moles of Ba(NO3)2 = 1.50 moles

Step 3: Calculate the total mass of the final solution

The total volume of the final solution is 1.00 L + 2.00 L = 3.00 L. Given that the density of the final solution is 2.00 g/mL, we can calculate the mass: Total mass = volume × density = \(3.00 L × 1000 \frac{mL}{L} × 2.00 \frac{g}{mL} = 6000 g\)

Step 4: Calculate the heat released or absorbed during the reaction

First, we find the temperature change: ΔT = final temperature − initial temperature ΔT = 42.0°C − 30.0°C = 12.0°C Next, we can calculate the heat released or absorbed using the formula: q = mass × specific heat capacity × ΔT = 6000 g × 6.37 J/g°C × 12.0°C = 458,664 J

Step 5: Calculate the enthalpy change per mole of BaSO4 formed

We can find the enthalpy change per mole of BaSO4 using the formula: Enthalpy change per mole = \( \frac{q}{moles\ of\ BaSO_{4}} \) Enthalpy change per mole = \( \frac{458,664 \mathrm{J}}{1.50\ \mathrm{moles}} \) = 305,776 J/mol The enthalpy change per mole of BaSO4 formed is 305,776 J/mol.

Key Concepts

Limiting Reactant

Understanding the concept of the limiting reactant is essential in chemistry because it determines the amount of product that can be formed in a chemical reaction. In a mixture of reactants, the limiting reactant is the one that is completely consumed first, and it limits the extent of the reaction. Once the limiting reactant is used up, the reaction stops, and no more product can form, even if other reactants are still present in excess.

To identify the limiting reactant, one must compare the mole ratio of the reactants present to the mole ratio required by the balanced chemical equation, just as it was done in the solution for Ba(NO3)2 and Na2SO4. The reactant that provides the lesser amount of moles required by the stoichiometry is the limiting reactant. This concept directly affects the calculation of theoretical yields in chemical reactions and is crucial for efficient resource usage in industrial processes.

Stoichiometry

The cornerstone of many chemistry calculations is stoichiometry. It involves using the relationships between reactants and products in a chemical reaction to determine quantities needed or produced. In essence, it's like a recipe that outlines the proportions of ingredients needed to make a particular product. Using stoichiometry, chemists can predict the amounts of substances consumed and created in a reaction.

For example, when baking a cake, you know that the right proportions of flour, sugar, and eggs are crucial. Likewise, in a chemical reaction, the balanced equation gives the proportion of reactants needed to form the desired products. In the provided solution, the balanced chemical equation dictates that one mole of Na2SO4 reacts with one mole of Ba(NO3)2 to produce one mole of BaSO4. Using these ratios, stoichiometry guides us in calculating how much product is formed from given amounts of reactants, which is fundamental in the calculation of the enthalpy change.

Calorimetry

The measurement of heat changes in physical and chemical processes is done using a technique called calorimetry. A device known as a calorimeter measures the amount of heat released or absorbed during these processes. In a chemical reaction, bonds are broken and formed, which involves energy changes that can be exothermic (giving off heat) or endothermic (absorbing heat).

Calorimetry applies the principle that the heat lost or gained by the surroundings is equal to the heat gained or lost by the system. The provided exercise involves a calorimetry calculation where the enthalpy change of a reaction is determined. By measuring the temperature change of the solution and knowing the specific heat capacity as well as the mass, you can calculate the total heat transfer. This value, divided by the number of moles of the product formed, gives the enthalpy change per mole, a quantity crucial for understanding energy requirements and transformations in chemical reactions.