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Chapter 3: Problem 42

At least \(25 \mu \mathrm{g}\) of tetrahydrocannabinol (THC), the active ingredient in marijuana, is required to produce intoxication. The molecular formula of THC is \(\mathrm{C}_{21} \mathrm{H}_{30} \mathrm{O}_{2}\). How many moles of THC does this \(25 \mu \mathrm{g}\) represent? How many molecules?

Short Answer

Expert verified
25 碌g of THC represents approximately \(7.95 \times 10^{-11}\) moles and \(4.79 \times 10^{13}\) molecules of THC.

Step by step solution

01

Calculate the molar mass of THC

First, we need to find the molar mass of THC. To do this, we'll use the molecular formula \(\mathrm{C}_{21} \mathrm{H}_{30} \mathrm{O}_{2}\) and the atomic masses of each element. We know the atomic masses of carbon (C), hydrogen (H), and oxygen (O) are approximately 12.01 g/mol, 1.01 g/mol, and 16.00 g/mol, respectively. So, the molar mass of THC is: Molar mass of THC = (21 脳 12.01 g/mol) + (30 脳 1.01 g/mol) + (2 脳 16.00 g/mol) = 314.47 g/mol
02

Convert the given mass of THC to moles

Next, we will convert the given mass of THC, \(25 \mu \mathrm{g}\), to moles. To do this, we will use the molar mass of THC (314.47 g/mol) and the conversion factor: 1 g = 1,000,000 碌g. Moles of THC = \(\frac{25 \mu \mathrm{g}}{314.47 \frac{\mathrm{g}}{\mathrm{mol}} \times 1,000,000 \frac{\mu \mathrm{g}}{\mathrm{g}}}\) = \(\frac{25}{314.47 \times 10^6}\) mol 鈮� 7.95 脳 10鈦宦孤� mol
03

Convert the number of moles to the number of molecules

Finally, we will convert the number of moles of THC (7.95 脳 10鈦宦孤� mol) to the number of molecules. To do this, we will use Avogadro's number (6.022 脳 10虏鲁 molecules/mol): Number of molecules = (7.95 脳 10鈦宦孤� mol) 脳 (6.022 脳 10虏鲁 molecules/mol) 鈮� 4.79 脳 10鹿鲁 molecules So, 25 碌g of THC represents approximately 7.95 脳 10鈦宦孤� moles and 4.79 脳 10鹿鲁 molecules of THC.

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Key Concepts

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

Molar Mass
Molar mass is a fundamental concept in chemistry that helps us understand how to relate the mass of a substance to the amount of that substance in moles. It is the mass of one mole of a given substance, typically expressed in grams per mole (g/mol). To calculate the molar mass, you need the molecular formula of the compound and the atomic masses of its elements.

In the example of tetrahydrocannabinol (THC), which has the molecular formula \(\mathrm{C}_{21} \mathrm{H}_{30} \mathrm{O}_{2}\), we find its molar mass by adding up the molar masses of each atom in the formula:
  • Carbon (C): 21 atoms 脳 12.01 g/mol = 252.21 g/mol
  • Hydrogen (H): 30 atoms 脳 1.01 g/mol = 30.30 g/mol
  • Oxygen (O): 2 atoms 脳 16.00 g/mol = 32.00 g/mol
When we sum these values, the molar mass of THC comes out to be 314.47 g/mol. This means that one mole of THC weighs 314.47 grams.
Avogadro's Number
Avogadro's number, which is 6.022 脳 10虏鲁, is a constant that denotes the number of particles in one mole of a substance. These particles can be atoms, molecules, ions, or other entities, depending on the context.

This number is immensely useful when converting between moles and the actual number of molecules or atoms. For instance, in our THC example, we calculated that the 25 micrograms of THC corresponds to 7.95 脳 10鈦宦孤� moles. Using Avogadro's number, we can further determine the number of THC molecules:
  • Number of molecules = 7.95 脳 10鈦宦孤� moles 脳 6.022 脳 10虏鲁 molecules/mole
  • This calculation results in approximately 4.79 脳 10鹿鲁 THC molecules
Understanding Avogadro's number is critical because it allows chemists to work on measurable scales, bridging the gap between atomic-scale and macro-scale quantities.
Molecular Formula
The molecular formula provides specific information about the number and types of atoms in a molecule, and it allows us to determine the molecular composition of a compound. The formula shows the exact number of each type of atom present in a molecule.

For THC, the molecular formula is \(\mathrm{C}_{21} \mathrm{H}_{30} \mathrm{O}_{2}\). This tells us that each molecule of THC is composed of:
  • 21 carbon atoms
  • 30 hydrogen atoms
  • 2 oxygen atoms
The specific composition in the molecular formula is crucial for determining both the chemical properties and the molar mass of the compound. By knowing the molecular formula, chemists can predict reactions and understand the structure and function of the molecule in various contexts. For THC, this understanding is essential to grasp its biological activity and how it interacts with the body.

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Most popular questions from this chapter

Write balanced chemical equations to correspond to each of the following descriptions: (a) When sulfur trioxide gas reacts with water, a solution of sulfuric acid forms. (b) Boron sulfide, \(\mathrm{B}_{2} \mathrm{~S}_{3}(s),\) reacts violently with water to form dissolved boric acid, \(\mathrm{H}_{3} \mathrm{BO}_{3},\) and hydrogen sulfide gas. (c) Phosphine, \(\mathrm{PH}_{3}(g)\), combusts in oxygen gas to form water vapor and solid tetraphosphorus decaoxide. (d) When solid mercury(II) nitrate is heated, it decomposes to form solid mercury(II) oxide, gaseous nitrogen dioxide, and oxygen. (e) Copper metal reacts with hot concentrated sulfuric acid solution to form aqueous copper(II) sulfate, sulfur dioxide gas, and water.

At least \(25 \mu \mathrm{g}\) of tetrahydrocannabinol \((\mathrm{THC}),\) the active ingredient in marijuana, is required to produce intoxication. The molecular formula of \(\mathrm{THC}\) is \(\mathrm{C}_{21} \mathrm{H}_{30} \mathrm{O}_{2}\). How many moles of THC does this 25 \mug represent? How many molecules? (b) Caffeine, a stimulant found in coffee, contains \(49.5 \% \mathrm{C}\), \(5.15 \% \mathrm{H}, 28.9 \% \mathrm{~N},\) and \(16.5 \% \mathrm{O}\) by mass and has a molar mass of \(195 \mathrm{~g} / \mathrm{mol}\). (c) Monosodium glutamate (MSG), a flavor enhancer in certain foods, contains \(35.51 \% \mathrm{C}, 4.77 \% \mathrm{H}, 37.85 \% \mathrm{O},\) \(8.29 \% \mathrm{~N},\) and \(13.60 \% \mathrm{Na},\) and has a molar mass of \(169 \mathrm{~g} / \mathrm{mol}\)

Write a balanced chemical equation for the reaction that occurs when (a) calcium metal undergoes a combination reaction with \(\mathrm{O}_{2}(g) ;\) (b) copper(II) hydroxide decomposes into copper(II) oxide and water when heated; (c) heptane, \(\mathrm{C}_{7} \mathrm{H}_{16}(l),\) burns in air; (d) methyl tert-butyl ether, \(\mathrm{C}_{5} \mathrm{H}_{12} \mathrm{O}(l)\) burns in air.

Aluminum hydroxide reacts with sulfuric acid as follows: $$ 2 \mathrm{Al}(\mathrm{OH})_{3}(s)+3 \mathrm{H}_{2} \mathrm{SO}_{4}(a q) \longrightarrow \mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3}(a q)+6 \mathrm{H}_{2} \mathrm{O}(l) $$ Which is the limiting reactant when \(0.500 \mathrm{~mol} \mathrm{Al}(\mathrm{OH})_{3}\) and \(0.500 \mathrm{~mol} \mathrm{H}_{2} \mathrm{SO}_{4}\) are allowed to react? How many moles of \(\mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3}\) can form under these conditions? How many moles of the excess reactant remain after the completion of the reaction?

Balance the following equations and indicate whether they are combination, decomposition, or combustion reactions: (a) \(\mathrm{C}_{3} \mathrm{H}_{6}(g)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g)+\mathrm{H}_{2} \mathrm{O}(g)\) (b) \(\mathrm{NH}_{4} \mathrm{NO}_{3}(s) \longrightarrow \mathrm{N}_{2} \mathrm{O}(g)+\mathrm{H}_{2} \mathrm{O}(g)\) (c) \(\mathrm{C}_{5} \mathrm{H}_{6} \mathrm{O}(l)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g)+\mathrm{H}_{2} \mathrm{O}(g)\) (d) \(\mathrm{N}_{2}(g)+\mathrm{H}_{2}(g) \longrightarrow \mathrm{NH}_{3}(g)\) (e) \(\mathrm{K}_{2} \mathrm{O}(s)+\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{KOH}(a q)\)
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