Problem 182 - Chemistry Matter and Change

Question

Chromium(III) nitrate forms a hydrate that is 40.50\(\%\) water by mass. What is its chemical formula?

Step-by-Step Solution

Answer

The chemical formula of the hydrate is \(Cr(NO_3)_3 \cdot 9H_2O\).

Step 1: Understanding Percent Composition

The hydrate consists of chromium(III) nitrate and water, which make up 40.50\(\%\) of its total mass. This means for every 100 g of the compound, 40.50 g is water.

Step 2: Use Molar Mass of Water

The molar mass of water \((H_2O)\) is approximately 18.01 g/mol. To find the moles of water in 40.50 g, we use the formula: \(\text{moles of } H_2O = \frac{40.50 \text{ g}}{18.01 \text{ g/mol}}\approx 2.25 \text{ mol}\) of water.

Step 3: Determine Remaining Mass for Chromium(III) Nitrate

Since 40.50 g of the 100 g of the compound is water, the remaining mass of the compound is \(100 - 40.50 = 59.50\text{ g}\), which is the mass of chromium(III) nitrate \((Cr(NO_3)_3)\).

Step 4: Calculate Moles of Chromium(III) Nitrate

The molar mass of chromium(III) nitrate \((Cr(NO_3)_3)\) is approximately \(238.02\text{ g/mol}\). The moles of chromium(III) nitrate are given by: \(\text{moles of } Cr(NO_3)_3 = \frac{59.50 \text{ g}}{238.02 \text{ g/mol}}\approx 0.250 \text{ mol}\).

Step 5: Determine the Ratio of Moles

The number of moles forms a ratio of \(2.25\) moles of water to \(0.25\) moles of \(Cr(NO_3)_3\). Simplifying this ratio, \(\frac{2.25}{0.25} = 9\). This means there are 9 moles of water for every mole of chromium(III) nitrate.

Step 6: Write the Chemical Formula

Given the ratio of moles, the chemical formula of the hydrate can be written as \(Cr(NO_3)_3 \cdot 9H_2O\).

Key Concepts

Percent Composition

When you think about a compound, it's made up of various elements, each contributing a certain amount to its total mass. Percent composition is a way to express how much of each component is present in the compound.

In the context of hydrate chemistry, a hydrate is a compound that includes water molecules within its structure. Here, the percent composition tells us what portion of the compound’s mass is due to water. For example, if a hydrate is 40.50% water, it means that out of every 100 grams of this compound, 40.50 grams is water.

Understanding the percent composition is crucial because it enables you to figure out how much of the compound isn't water, which helps in identifying the chemical formula of the core compound. Once you have that information, you can also find out the rest of the compound's composition, which is often the anhydrous salt.

Molar Mass Calculation

The molar mass of a compound is essential because it describes the mass of one mole of that substance, and it allows us to convert between mass and number of moles very easily.

To calculate the molar mass, you sum up the atomic masses of all atoms in the compound's chemical formula. For instance, the molar mass of water ( $H_2O$ ) is calculated by adding the atomic masses of 2 hydrogen ( $H$ ) atoms and 1 oxygen ( $O$ ) atom, resulting in approximately 18.01 g/mol.

Why is this significant in our problem? Knowing the molar mass of water lets us convert the mass of water (from the percent composition) into moles. This provides a quantitative measure needed to form the ratio between the anhydrous part and the water within the hydrate. This is an important step in determining the formula of the hydrate compound accurately.

Chemical Formula Determination

Once you've calculated the molar masses and converted the masses into moles, the next step is to determine the chemical formula of the hydrate.

The chemical formula of a hydrate typically looks like \(X \cdot nH_2O\), where \(X\) is the anhydrous salt and \(n\) is the number of water molecules per formula unit of \(X\).
To find this ratio, we calculate the moles of water and the moles of the anhydrous part of the compound. For instance, if you're given the compound chromium(III) nitrate with water, and you've calculated that you have 2.25 moles of water and 0.25 moles of chromium(III) nitrate, the ratio is 2.25:0.25.
By simplifying the mole ratio (in this case, 9:1), you can determine the chemical formula of the hydrate as \(Cr(NO_3)_3 \cdot 9H_2O\), indicating that there are 9 water molecules accompanying each formula unit of chromium(III) nitrate. This is how these calculations and understanding come full circle to give us the complete formula of the compound.