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Chapter 3: Problem 39

The molecular formula of saccharin, an artificial sweetener, is \(\mathrm{C}_{7} \mathrm{H}_{5} \mathrm{NO}_{3} \mathrm{~S} .(\mathbf{a})\) What is the molar mass of saccharin? (b) How many moles of sachharin are in $2.00 \mathrm{mg}\( of this substance? (c) How many molecules are in \)2.00 \mathrm{mg}\( of this substance? (d) How many C atoms are present in \)2.00 \mathrm{mg}$ of saccharin?

Short Answer

Expert verified
(a) The molar mass of saccharin is approximately 183.18 g/mol. (b) In 2.00 mg of saccharin, there are about 1.09 × 10^{-5} moles. (c) In 2.00 mg of saccharin, there are approximately 6.56 × 10^{18} molecules. (d) In 2.00 mg of saccharin, there are approximately 4.59 × 10^{19} carbon atoms.

Step by step solution

01

Calculate the molar mass of saccharin

To calculate the molar mass of saccharin, we'll need to find the molar mass of each element in the molecule and add them together. The molecular formula of saccharin is \(\mathrm{C}_{7} \mathrm{H}_{5} \mathrm{NO}_{3} \mathrm{~S}\), so its molar mass will be: Molar mass of saccharin = (7 × molar mass of C) + (5 × molar mass of H) + (1 × molar mass of N) + (3 × molar mass of O) + (1 × molar mass of S) Using the molar masses of each element (C: 12.01 g/mol, H: 1.01 g/mol, N: 14.01 g/mol, O: 16.00 g/mol, S: 32.07 g/mol): Molar mass of saccharin ≈ (7 × 12.01) + (5 × 1.01) + (1 × 14.01) + (3 × 16.00) + (1 × 32.07) Molar mass of saccharin ≈ 183.18 g/mol
02

Calculate the number of moles of saccharin

To find the number of moles of saccharin in 2.00 mg, we'll use the molar mass we calculated and convert the mass to grams: 2.00 mg = 0.00200 g Number of moles = (mass of substance) / (molar mass of substance) Number of moles of saccharin = 0.00200 g / 183.18 g/mol ≈ 1.09 × 10^{-5} moles
03

Calculate the number of molecules of saccharin

To find the number of molecules in the sample, we'll use Avogadro's number (6.022 × 10^{23} molecules/mol): Number of molecules = (number of moles) × (Avogadro's number) Number of molecules of saccharin ≈ 1.09 × 10^{-5} moles × 6.022 × 10^{23} molecules/mol ≈ 6.56 × 10^{18} molecules
04

Calculate the number of C atoms in 2.00 mg of saccharin

To find the number of carbon atoms in 2.00 mg of saccharin, first find the number of carbon atoms in one molecule of saccharin: C atoms in one molecule of saccharin = 7 (from the molecular formula \(\mathrm{C}_{7} \mathrm{H}_{5} \mathrm{NO}_{3} \mathrm{~S}\)) Next, multiply the number of carbon atoms in one molecule by the total number of molecules: C atoms in 2.00 mg of saccharin = (number of C atoms per molecule) × (number of molecules) C atoms in 2.00 mg of saccharin ≈ 7 × 6.56 × 10^{18} ≈ 4.59 × 10^{19} carbon atoms

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