Problem 35 - Chemistry The Central Science In Si Units

Question

Calculate the following quantities: (a) mass, in grams, of 0.105 mol sucrose \(\left(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right)\) (b) moles of \(\mathrm{Zn}\left(\mathrm{NO}_{3}\right)_{2}\) in \(143.50 \mathrm{~g}\) of this substance (c) number of molecules in \(1.0 \times 10^{-6} \mathrm{~mol} \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}\) (d) number of \(\mathrm{N}\) atoms in \(0.410 \mathrm{~mol} \mathrm{NH}_{3}\)

Step-by-Step Solution

Answer

(a) 35.94 g sucrose; (b) 0.757 mol Zn(NO3)2; (c) 6.022 × 10^17 molecules of C2H5OH; (d) 2.47 × 10^23 N atoms in NH3.

Step 1: Calculate Molar Mass (a)

Identify the molar mass of sucrose \ \( \text{C}_{12}\text{H}_{22}\text{O}_{11} \). Calculate by adding the atomic masses of 12 carbon atoms, 22 hydrogen atoms, and 11 oxygen atoms: - Carbon: \(12 \times 12.01 = 144.12\) g/mol- Hydrogen: \(22 \times 1.008 = 22.176\) g/mol- Oxygen: \(11 \times 16.00 = 176.00\) g/molAdd them together: \[ \text{Molar mass of sucrose} = 342.30 \text{ g/mol} \]

Step 2: Mass of Sucrose (a)

To find the mass in grams of 0.105 mol sucrose, use the formula:\[ \text{Mass} = \text{moles} \times \text{molar mass} \]\[ \text{Mass} = 0.105 \text{ mol} \times 342.30 \text{ g/mol} = 35.94 \text{ g} \]

Step 3: Calculate Molar Mass of Zn(NO3)2 (b)

Identify the molar mass of \( \text{Zn(NO}_3)_2 \):- Zinc: \(1 \times 65.38 = 65.38\) g/mol- Nitrogen: \(2 \times 14.01 = 28.02\) g/mol- Oxygen: \(6 \times 16.00 = 96.00\) g/molAdd them together:\[ \text{Molar mass of } \text{Zn(NO}_3)_2 = 189.40 \text{ g/mol} \]

Step 4: Moles of Zn(NO3)2 (b)

To find the moles of \( \text{Zn(NO}_3)_2 \) in 143.50 g, use the formula:\[ \text{Moles} = \frac{\text{Mass}}{\text{Molar mass}} \]\[ \text{Moles} = \frac{143.50 \text{ g}}{189.40 \text{ g/mol}} \approx 0.757 \text{ mol} \]

Step 5: Calculate Molecules of C2H5OH (c)

Start with \(1.0 \times 10^{-6} \text{ mol} \) of \( \text{C}_2\text{H}_5\text{OH} \). Use Avogadro's number, \(6.022 \times 10^{23} \text{ molecules/mol} \):\[ \text{number of molecules} = \text{moles} \times \text{Avogadro's number} \]\[ \text{number of molecules} = 1.0 \times 10^{-6} \times 6.022 \times 10^{23} \]\[ \text{number of molecules} \approx 6.022 \times 10^{17} \]

Step 6: Calculate Nitrogen Atoms in NH3 (d)

Start with \(0.410 \text{ mol} \text{ NH}_3\). Each molecule contains 1 nitrogen atom. Use Avogadro's number to find the total:\[ \text{number of N atoms} = \text{moles} \times \text{Avogadro's number} \]\[ \text{number of N atoms} = 0.410 \times 6.022 \times 10^{23} \]\[ \text{number of N atoms} \approx 2.47 \times 10^{23} \]

Key Concepts

Understanding Molar Mass Calculation

The molar mass of a substance is a key concept in chemistry that helps us bridge the gap between the microscopic world of atoms and molecules and the macroscopic world we can observe and measure. It gives us a way to convert between the amount of substance in moles and its mass in grams. To calculate the molar mass, sum up the atomic masses of all the atoms in a molecule:

• Identify each type of atom in the molecule.
• Multiply the atomic mass of each element by the number of atoms of that element in the molecule.
• Add the masses of all the elements together to get the total molar mass.

For example, sucrose (C₁₂H₂₂O₁₁) requires adding the atomic masses of 12 carbon atoms, 22 hydrogen atoms, and 11 oxygen atoms. Carbon has an atomic mass of 12.01 g/mol, hydrogen has 1.008 g/mol, and oxygen has 16.00 g/mol. When summed, these values provide the molar mass of sucrose as 342.30 g/mol.

Exploring Avogadro's Number

Avogadro's number, approximately 6.022 x 10²³, is a fundamental constant in chemistry. It connects the amount of substance to the number of atoms or molecules present. Understanding Avogadro's number is crucial for converting between moles and molecules:

• One mole of any substance contains exactly 6.022 x 10²³ entities (atoms, molecules, etc.).
• This immense number helps chemists work with quantities at the macroscopic scale, making it easier to handle bulk substances and perform precise calculations.

For instance, if you have 1.0 x 10^-6 moles of ethanol (CH₃CH₂OH), using Avogadro's number will allow you to find out the exact number of ethanol molecules present. By multiplying the number of moles by Avogadro's number, you get approximately 6.022 x 10¹⁷ molecules.

Mastering Mass-to-Mole Conversion

Converting between mass and moles is a frequently used skill in chemistry that allows you to relate how much of a substance you have to how many moles it is, or vice versa. To perform a mass-to-mole conversion or its reverse, you'll apply these straightforward steps:

• Determine the molar mass of the substance from the periodic table or a reference.
• To find moles: Divide the mass of the substance (in grams) by its molar mass (g/mol).
• To find mass from moles: Multiply the number of moles by the molar mass.

For example, let's calculate the moles of zinc nitrate (Zn(NO₃)₂) given 143.50 g of it. With a molar mass of 189.40 g/mol for zinc nitrate, dividing the mass by the molar mass yields approximately 0.757 moles.