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Chapter 3: Problem 149

Vitamin A has a molar mass of \(286.4 \mathrm{~g} / \mathrm{mol}\) and a general molecular formula of \(\mathrm{C}_{x} \mathrm{H}_{2} \mathrm{E}\), where \(\mathrm{E}\) is an unknown element. If vitamin \(\mathrm{A}\) is \(83.86 \% \mathrm{C}\) and \(10.56 \% \mathrm{H}\) by mass, what is the molecular formula of vitamin \(\mathrm{A}\) ?

Short Answer

Expert verified
The molecular formula of vitamin A is \( \mathrm{C}_{20} \mathrm{H}_{30} \mathrm{O}\).

Step by step solution

01

Determine moles of carbon and hydrogen in one mole of vitamin A

Since vitamin A is \(83.86 \% \: \mathrm{C}\) and \(10.56 \% \: \mathrm{H}\) by mass, you can find the mass of carbon and hydrogen in one mole of vitamin A as follows: Mass of carbon: \(0.8386 \times 286.4 \: \mathrm{g/mol} = 240 \: \mathrm{g/mol}\) Mass of hydrogen: \(0.1056 \times 286.4 \: \mathrm{g/mol} = 30.24 \: \mathrm{g/mol}\) Next, find the moles of carbon and hydrogen in one mole of vitamin A: Moles of carbon: \(\frac{240 \: \mathrm{g/mol}}{12.01 \: \mathrm{g/mol}} = 20\) Moles of hydrogen: \(\frac{30.24 \: \mathrm{g/mol}}{1.008 \: \mathrm{g/mol}} = 30\) The value of x in the general molecular formula of vitamin A is 20.
02

Determine the mass of the unknown element E

Now, subtract the sum of the mass of carbon and hydrogen from the molar mass of vitamin A to find the mass of the unknown element E: Mass of E: \(286.4 \: \mathrm{g/mol} - 240 \: \mathrm{g/mol} - 30.24 \: \mathrm{g/mol} = 16.16 \: \mathrm{g/mol}\)
03

Identify the unknown element E and its moles

Based on the mass of the unknown element E, it can be identified as oxygen (O) with a molar mass of approximately \(16 \: \mathrm{g/mol}\). Now, find the moles of E (oxygen) in one mole of vitamin A: Moles of oxygen: \(\frac{16.16 \: \mathrm{g/mol}}{16 \: \mathrm{g/mol}} = 1\)
04

Write the molecular formula of vitamin A

Using the moles of carbon, hydrogen, and oxygen, the molecular formula of vitamin A can be written as: Vitamin A: \( \mathrm{C}_{20} \mathrm{H}_{30} \mathrm{O}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Molar Mass
The molar mass is a fundamental aspect of chemistry, representing the mass of one mole of a substance. In simpler terms, it is the weight of one mole of atoms, molecules, or compounds. A mole is a specific number of particles, namely 6.022 x 10^23 particles, which is Avogadro's number. To find the molar mass, you sum up the atomic masses of all the atoms in the molecule.

For instance, in the case of vitamin A with a molar mass of 286.4 g/mol, that number signifies the weight of 6.022 x 10^23 molecules of vitamin A. The molar mass is critical when converting between the mass of a material and the number of moles, as seen in the original exercise where the mass percentages were converted to moles to determine the molecular formula.
Percent Composition
Understanding percent composition is crucial when analyzing the makeup of compounds. It indicates the mass percentage of each element within a compound. By knowing the percent composition, you can determine how much of each element, by mass, is present in a mole of the compound. If we take our vitamin A example, the exercise provides that it is 83.86% carbon and 10.56% hydrogen by mass.

These values help us deduce the molecular formula of the compound by calculating the proportion of the elements in a mole. Even if the full molecular formula is unknown, percent composition allows chemists to start piecing together what the compound may consist of, as reflected in the step-by-step solution provided for calculating the moles of carbon and hydrogen.
Mole Concept
The mole concept is a bridge between the microscopic world of atoms and molecules and the macroscopic world we observe. It allows chemists to count particles by weighing them. When we say one mole of a substance, like in the exercise's context, we're talking about having Avogadro's number of molecules or atoms of that substance.

Within the realm of molecular formula determination, the mole concept lets us move from the mass of an element (in grams) in a compound to the number of moles of that element, as shown when the moles of carbon and hydrogen were determined in the exercise. Working through the problem, understanding the mole concept is essential in deducing the accurate molecular formula, as the number of moles directly reflects the subscript numbers in the formula for each element.

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Most popular questions from this chapter

The aspirin substitute, acetaminophen \(\left(\mathrm{C}_{8} \mathrm{H}_{9} \mathrm{O}_{2} \mathrm{~N}\right)\), is produced by the following three-step synthesis: I. \(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{3} \mathrm{~N}(s)+3 \mathrm{H}_{2}(g)+\mathrm{HCl}(a q) \longrightarrow\) $$ \mathrm{C}_{6} \mathrm{H}_{3} \mathrm{ONCl}(s)+2 \mathrm{H}_{2} \mathrm{O}(t) $$ II. \(\mathrm{C}_{6} \mathrm{H}_{8} \mathrm{ONCl}(s)+\mathrm{NaOH}(a q) \longrightarrow\) $$ \mathrm{C}_{6} \mathrm{H}_{7} \mathrm{ON}(s)+\mathrm{H}_{2} \mathrm{O}(t)+\mathrm{NaCl}(a q) $$ III. \(\mathrm{C}_{6} \mathrm{H}_{7} \mathrm{ON}(s)+\mathrm{C}_{4} \mathrm{H}_{6} \mathrm{O}_{3}(l) \longrightarrow\) $$ \mathrm{C}_{8} \mathrm{H}_{9} \mathrm{O}_{2} \mathrm{~N}(s)+\mathrm{HC}_{2} \mathrm{H}_{3} \mathrm{O}_{2}(l) $$ The first two reactions have percent yields of \(87 \%\) and \(98 \%\) by mass, respectively. The overall reaction yields 3 moles of acetaminophen product for every 4 moles of \(\mathrm{C}_{6} \mathrm{H}_{3} \mathrm{O}_{3} \mathrm{~N}\) reacted. a. What is the percent yield by mass for the overall process? b. What is the percent yield by mass of Step III?

Coke is an impure form of carbon that is often used in the industrial production of metals from their oxides. If a sample of coke is \(95 \%\) carbon by mass, determine the mass of coke needed to react completely with \(1.0\) ton of copper(II) oxide. $$ 2 \mathrm{CuO}(s)+\mathrm{C}(s) \longrightarrow 2 \mathrm{Cu}(s)+\mathrm{CO}_{2}(g) $$

With the advent of techniques such as scanning tunneling microscopy, it is now possible to "write" with individual atoms by manipulating and arranging atoms on an atomic surface. a. If an image is prepared by manipulating iron atoms and their total mass is \(1.05 \times 10^{-20} \mathrm{~g}\), what number of iron atoms were used? b. If the image is prepared on a platinum surface that is exactly 20 platinum atoms high and 14 platinum atoms wide, what is the mass (grams) of the atomic surface? c. If the atomic surface were changed to ruthenium atoms and the same surface mass as determined in part b is used, what number of ruthenium atoms is needed to construct the surface?

Bornite \(\left(\mathrm{Cu}_{3} \mathrm{FeS}_{3}\right)\) is a copper ore used in the production of copper. When heated, the following reaction occurs: \(2 \mathrm{Cu}_{3} \mathrm{FeS}_{3}(s)+7 \mathrm{O}_{2}(g) \longrightarrow 6 \mathrm{Cu}(s)+2 \mathrm{FeO}(s)+6 \mathrm{SO}_{2}(g)\) If \(2.50\) metric tons of bornite is reacted with excess \(\mathrm{O}_{2}\) and the process has an \(86.3 \%\) yield of copper, what mass of copper is produced?

What number of atoms of nitrogen are present in \(5.00 \mathrm{~g}\) of each of the following? a. glycine, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{O}_{2} \mathrm{~N}\) b. magnesium nitride c. calcium nitrate d. dinitrogen tetroxide
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