Question

Narceine is a narcotic in opium that crystallizes from solution as a hydrate that contains 10.8 mass \(\%\) water and has a molar mass of \(499.52 \mathrm{~g} / \mathrm{mol} .\) Determine \(x\) in narceine \(x \mathrm{H}_{2} \mathrm{O}\).

Short answer

The value of x in narceine \( x \mathrm{H}_2 \mathrm{O} \) is 3.

Step-by-step solution

Calculate the mass of water in the hydrate

Since the hydrate is 10.8% water by mass, the mass of water in one mole of the hydrate can be calculated as: \[ \text{Mass of water} = \frac{10.8}{100} \times 499.52 \text{ g/mol} = 53.95 \text{ g/mol} \]

Determine the molar mass of water

The molar mass of water (H\textsubscript{2}O) is determined by adding the molar masses of 2 hydrogen atoms and 1 oxygen atom: \[ \text{Molar mass of } H_2O = 2 \times 1.01 + 16.00 = 18.02 \text{ g/mol} \]

Calculate the number of moles of water in the hydrate

Use the mass of water calculated in Step 1 and the molar mass of water from Step 2 to determine the number of moles of water: \[ x = \frac{53.95 \text{ g/mol}}{18.02 \text{ g/mol}} \approx 2.99 \approx 3 \]

Conclude the value of x

Since the calculation in Step 3 approximates to 3, the value of x in narceine \( x \mathrm{H}_2 \mathrm{O} \) is 3.

Key concepts

Molar Mass

The molar mass is the mass of one mole of a substance. It is expressed in grams per mole (g/mol). To find the molar mass of a compound, add up the molar masses of all the atoms in the formula. For instance, water (H₂O) has a molar mass calculated as follows:

Take the molar masses of 2 hydrogen atoms (2 × 1.01 g/mol) and 1 oxygen atom (16.00 g/mol).
Add them up: 2.02 g/mol + 16.00 g/mol = 18.02 g/mol.

Knowing a compound's molar mass is essential for various calculations in chemistry, including determining the number of moles in a sample.

Hydration

Hydration refers to the inclusion of water molecules in the structure of a compound. These compounds are called hydrates. When a substance crystallizes from water, it often traps water molecules as part of its solid form. The water content in hydrates is usually expressed in terms of mass percent or mole ratios.

In this exercise, narceine crystallizes as a hydrate containing 10.8% water by mass. That means that for every 100 grams of hydrated narceine, there are 10.8 grams of water. Knowing the percentage of water helps in determining the number of water molecules associated with one formula unit of the compound.

Narcotics in Opium

Opium contains several narcotic substances, including narceine. Narcotics are compounds that can relieve pain and induce sleep. Narceine is one such compound that crystallizes from solution as a hydrate, forming crystals that include water molecules as part of their structure.

Extracting and understanding these substances is important in pharmacology and medicine. The hydration properties of these narcotic compounds can influence their stability and how they are processed and used medicinally.

Mass Percent Composition

Mass percent composition tells us the percentage of each component in a compound by mass. It is useful in determining the proportion of each element or group of atoms, such as water in a hydrate.

To find the mass percent of water in narceine hydrate:

• Calculate the mass of the entire compound.
• Determine the mass of the water portion.
• Express the water mass as a percentage of the total mass.

In this problem, narceine has 10.8 mass percent water. Calculating the exact water mass per mole and correlating it to the total molar mass helps in determining the hydration number, which was found to be 3 (i.e., narceine contains 3 water molecules per formula unit).