Problem 144 - Chemistry The Molecular Science

Question

Through a series of reactions, a 12.3-g sample of potassium carbonate was chemically reacted so that all of its carbon was found in \(\mathrm{K}_{2} \mathrm{Zn}_{3}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]_{2} .\) Calculate the mass of \(\mathrm{K}_{2} \mathrm{Zn}_{3}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]_{2}\) formed.

Step-by-Step Solution

Answer

5.18 g of K2Zn3[Fe(CN)6]2 is formed.

Step 1: Calculate moles of Potassium Carbonate

First, calculate the molar mass of potassium carbonate (\(K_2CO_3\): 2 potassium (K), 1 carbon (C), and 3 oxygen (O). \[\text{Molar mass of } K_2CO_3 = 2(39.10) + 12.01 + 3(16.00) = 138.21 \text{ g/mol}\]Using the mass of the sample, calculate moles: \[\text{Moles of } K_2CO_3 = \frac{12.3 \text{ g}}{138.21 \text{ g/mol}} = 0.089 \text{ mol}\]

Step 2: Calculate moles of Carbon in Potassium Carbonate

Potassium carbonate (\(K_2CO_3\)) contains one mole of carbon per mole of compound. Therefore, the moles of carbon are the same as the moles of Potassium Carbonate:\[\text{Moles of Carbon} = 0.089 \text{ mol}\]

Step 3: Calculate moles of K2Zn3[Fe(CN)6]2 formed

Assuming all the carbon atoms from potassium carbonate are transferred into \(K_2Zn_3[Fe(CN)_6]_2\), we need to consider the stoichiometry of carbon in \(K_2Zn_3[Fe(CN)_6]_2\).Each \([Fe(CN)_6]^{4-}\) contains 6 moles of cyanide ions (CN) and therefore 6 moles of carbon per mole of \([Fe(CN)_6]^{4-}\). Since \(K_2Zn_3[Fe(CN)_6]_2\) contains 2 \([Fe(CN)_6]^{4-}\) complexes per formula unit, each molecule has:\[6 \times 2 = 12 \text{ moles of carbon} \] Using stoichiometry:\[\text{Moles of } K_2Zn_3[Fe(CN)_6]_2 = \frac{\text{moles of Carbon}}{12} = \frac{0.089 \text{ mol}}{12} = 0.00742 \text{ mol}\]

Step 4: Calculate Mass of K2Zn3[Fe(CN)6]2

To find the mass formed, calculate the molar mass of \(K_2Zn_3[Fe(CN)_6]_2\):\[2K = 2(39.10), 3Zn = 3(65.38), 2Fe = 2(55.85), 12C = 12(12.01), 12N = 12(14.01)\]Adding up the molar masses:\[2(39.10) + 3(65.38) + 2(55.85) + 12(12.01) + 12(14.01) = 698.41 \text{ g/mol}\]Now, calculate the mass:\[\text{Mass of } K_2Zn_3[Fe(CN)_6]_2 = 0.00742 \text{ mol} \times 698.41 \text{ g/mol} = 5.18 \text{ g}\]

Key Concepts

Molar Mass Calculation

Molar mass is a crucial concept in chemistry that helps us convert between grams and moles, allowing us to engage with chemical reactions quantitatively. Molar mass is essentially the sum of the atomic masses of all atoms in a molecule. To calculate the molar mass, consider the periodic table values of each element involved in the molecule. For instance, to determine the molar mass of potassium carbonate \(K_2CO_3\), follow these steps:

• Locate the atomic masses: Potassium \(K\) has an atomic mass of approximately 39.10 g/mol, Carbon \(C\) is 12.01 g/mol, and Oxygen \(O\) is around 16.00 g/mol.
• Multiply the atomic mass by the number of atoms of each element in the compound: 2 Potassium atoms give \([2 \times 39.10]\), 1 Carbon atom gives \(12.01\), and 3 Oxygen atoms give \([3 \times 16.00]\).
• Add these together to get the molar mass: \( ext{Molar mass of } K_2CO_3 = 138.21 ext{ g/mol}\).

Understanding the molar mass allows you to calculate moles from a given mass and vice versa, which is vital for stoichiometric calculations.

Chemical Reactions

Chemical reactions involve the transformation of substances through the breaking and forming of chemical bonds, resulting in new substances with different properties. In the provided exercise, chemical reactions are key to transforming potassium carbonate into \(K_2Zn_3[Fe(CN)_6]_2\). Here's what happens during a reaction:

• Reactants, the starting substances, undergo chemical changes to form products, the new substances.
• These reactions must obey the law of conservation of mass, meaning atoms are neither created nor destroyed. Only their arrangements change.
• The stoichiometry of a reaction, which entails the quantitative relationships between reactants and products, guides these transformations.
• In the case of transferring all carbon atoms from potassium carbonate to \(K_2Zn_3[Fe(CN)_6]_2\), each formula unit must precisely match the carbon atoms from the reactant to the product.

Mastering chemical reactions allows you to predict product formation and identify the necessary quantities of reactants needed.

Moles Conversion

Conversions between moles and other units, such as grams or molecules, are fundamental in stoichiometry, helping chemists quantify substances in a reaction. This exercise requires understanding how to convert the mass of potassium carbonate into moles, which then informs product creation in \(K_2Zn_3[Fe(CN)_6]_2\). Here's how to perform conversions:

• To convert grams to moles, use the relationship \( ext{moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}}\).
• Using the given mass of 12.3 grams of \(K_2CO_3\) and its calculated molar mass of 138.21 g/mol, find the moles \( \approx 0.089 ext{ mol}\).
• To further convert to the product \(K_2Zn_3[Fe(CN)_6]_2\), recognize that the moles of carbon in the reactant and product must be equivalent, leading to division by the stoichiometry of carbon (12 moles per product unit) to find \( \approx 0.00742 ext{ mol}\) of product formed.
• This final step allows calculation of the product's mass using its molar mass.

By grasping mole conversions, you can manage and predict various amounts of substances engaged in chemical reactions.