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At least \(25 \mu \mathrm{g}\) of tetrahydrocannabinol (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 \(25 \mu \mathrm{g}\) represent? How many molecules?

Short Answer

Expert verified
In 25 碌g of tetrahydrocannabinol (THC), with a molecular formula of C鈧傗倎H鈧冣個O鈧, there are approximately 7.95 x 10鈦烩伕 moles and 4.79 x 10鹿鈦 molecules.

Step by step solution

01

Find the molar mass of THC

To find the molar mass of THC (C鈧傗倎H鈧冣個O鈧), we can add up the molar mass of each of the individual elements, multiplied by the number of atoms of that element in the molecule: Molar mass of THC = (21 x Molar mass of carbon) + (30 x Molar mass of hydrogen) + (2 x Molar mass of oxygen) Using the periodic table, the molar mass of carbon is 12.01 g/mol, hydrogen is 1.008 g/mol, and oxygen is 16 g/mol: Molar mass of THC = (21 x 12.01 g/mol) + (30 x 1.008 g/mol) + (2 x 16 g/mol) = 314.47 g/mol.
02

Convert micrograms to grams

Since the given mass is in micrograms, we need to convert it to grams, as the molar mass is in grams per mole. 25 碌g = 25 x 10鈦烩伓 grams
03

Find the number of moles in 25 碌g of THC

Now that we have the molar mass of THC and the mass in grams, we can find the number of moles using the formula: moles = mass / molar mass moles of THC = (25 x 10鈦烩伓 g) / (314.47 g/mol) = 7.95 x 10鈦烩伕 mol
04

Find the number of molecules in the given number of moles

Now, we can use Avogadro's number (6.022 x 10虏鲁 molecules/mole) to find the number of molecules in these moles: Number of molecules = moles x Avogadro's number Number of molecules of THC = (7.95 x 10鈦烩伕 mol) x (6.022 x 10虏鲁 molecules/mol) = 4.79 x 10鹿鈦 molecules So, in 25 碌g of THC, there are approximately 4.79 x 10鹿鈦 molecules of tetrahydrocannabinol.

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Key Concepts

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

Chemical Formulas
Chemical formulas are crucial in chemistry as they provide a shorthand way of representing the composition of molecules and compounds. These formulas indicate the type and number of atoms in a molecule. For example, in the original exercise, the chemical formula of tetrahydrocannabinol (THC) is \(\text{C}_{21}\text{H}_{30}\text{O}_{2}\). This illustrates that each molecule of THC consists of 21 carbon atoms, 30 hydrogen atoms, and 2 oxygen atoms.
The subscripts in a chemical formula convey the moles of each element in one mole of the compound. By understanding chemical formulas, scientists can easily determine the proportions of different elements in a substance and calculate properties like molar mass. Recognizing these formulas is essential for performing further calculations based on the mole concept or other chemical principles.
Mole Concept
The mole concept is a fundamental principle in chemistry that allows chemists to count particles, such as atoms and molecules, using a measurable quantity. One mole of any substance contains the same number of particles, which facilitates calculations between the mass and number of atoms in a sample.
The mole concept ties the macroscopic world of grams and liters to the microscopic world of atoms and molecules. By using the formula \(\text{moles} = \frac{\text{mass}}{\text{molar mass}}\), we can determine the number of moles of a substance when given the mass and the molar mass. In the exercise, converting 25 碌g of THC into moles helps us transition from a tiny mass into an understandable quantity on the mole scale. This allows us to further compute how many molecules we are considering.
Avogadro's Number
Avogadro's number is a key constant in chemistry, defined as \(6.022 \times 10^{23}\) particles per mole. It provides the link between the macroscopic scale (that we can measure) and the atomic scale (which we cannot directly observe).
For instance, once we know the number of moles of THC, Avogadro's number allows us to calculate the exact number of molecules in that amount. By multiplying the moles obtained from the mass by Avogadro's number \((\text{moles} \times 6.022 \times 10^{23})\), students can transition from grams to individual molecules. This tool is invaluable for understanding reactions as it bridges the gap between the scale of laboratory measurements and molecular interactions.

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Most popular questions from this chapter

(a) What is the mass, in grams, of a mole of \({ }^{12} \mathrm{C}\) ? (b) How many carbon atoms are present in a mole of \({ }^{12} \mathrm{C}\) ?

Write balanced chemical equations to correspond to each of the following descriptions: (a) Solid calcium carbide, \(\mathrm{CaC}_{2}\), reacts with water to form an aqueous solution of calcium hydroxide and acetylene gas, \(\mathrm{C}_{2} \mathrm{H}_{2}\). (b) When solid potassium chlorate is heated, it decomposes to form solid potassium chloride and oxygen gas. (c) Solid zinc metal reacts with sulfuric acid to form hydrogen gas and an aqueous solution of zinc sulfate. (d) When liquid phosphorus trichloride is added to water, it reacts to form aqueous phosphorous acid, \(\mathrm{H}_{3} \mathrm{PO}_{3}(a q)\), and aqueous hydrochloric acid. (e) When hydrogen sulfide gas is passed over solid hot iron(III) hydroxide, the resultant reaction produces solid iron(III) sulfide and gaseous water.

(a) What scientific principle or law is used in the process of balancing chemical equations? (b) In balancing equations, why should you not change subscripts in chemical formulas? (c) How would one write out liquid water, water vapor, aqueous sodium chloride, and solid sodium chloride in chemical equations?

Very small crystals composed of 1000 to 100,000 atoms, called quantum dots, are being investigated for use in electronic devices. (a) A quantum dot was made of solid silicon in the shape of a sphere, with a diameter of \(4 \mathrm{~nm}\). Calculate the mass of the quantum dot, using the density of silicon \(\left(2.3 \mathrm{~g} / \mathrm{cm}^{3}\right)\) (b) How many silicon atoms are in the quantum dot? (c) The density of germanium is \(5.325 \mathrm{~g} / \mathrm{cm}^{3} .\) If you made a \(4 \mathrm{~nm}\) quantum dot of germanium, how many Ge atoms would it contain? Assume the dot is spherical.

Calculate the percentage by mass of the indicated element in the following compounds: (a) carbon in acetylene, \(\mathrm{C}_{2} \mathrm{H}_{2}\), a gas used in welding; (b) hydrogen in ascorbic acid, \(\mathrm{HC}_{6} \mathrm{H}_{7} \mathrm{O}_{6}\), also known as vitamin \(\mathrm{C}\); (c) hydrogen in ammonium sulfate, \(\left(\mathrm{NH}_{4}\right)_{2} \mathrm{SO}_{4}\), a substance used as a nitrogen fertilizer; (d) platinum in \(\mathrm{PtCl}_{2}\left(\mathrm{NH}_{3}\right)_{2}\), a chemotherapy agent called cisplatin; (e) oxygen in the female sex hormone estradiol, \(\mathrm{C}_{18} \mathrm{H}_{24} \mathrm{O}_{2} ;\) (f) carbon in capsaicin, \(\mathrm{C}_{18} \mathrm{H}_{27} \mathrm{NO}_{3}\), the com- pound that gives the hot taste to chili peppers.

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