Problem 72 - General Chemistry: Principles and Modern Applications

Question

Anhydrous sodium sulfate, \(\mathrm{Na}_{2} \mathrm{SO}_{4},\) absorbs water vapor and is converted to the decahydrate, \(\mathrm{Na}_{2} \mathrm{SO}_{4} \cdot 10 \mathrm{H}_{2} \mathrm{O} .\) How much would the mass of \(24.05 \mathrm{g}\) of anhydrous \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) increase if converted completely to the decahydrate?

Step-by-Step Solution

Answer

The mass of anhydrous sodium sulfate would increase by approximately 30.4 g if completely converted to the decahydrate.

Step 1: Calculate the Molar Masses

Start by calculating the molar mass of both anhydrous \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) and its decahydrate form \(\mathrm{Na}_{2} \mathrm{SO}_{4} \cdot 10 \mathrm{H}_{2} \mathrm{O}\). The molar mass of \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) is approx. 142.04 g/mol, derived by adding the molar masses of: 2 Sodium atoms (2*23.0 g/mol), 1 Sulfur atom (32.1 g/mol), and 4 Oxygen atoms (4*16.0 g/mol). On the other hand, the molar mass of \(\mathrm{Na}_{2} \mathrm{SO}_{4} \cdot 10 \mathrm{H}_{2} \mathrm{O}\) is approx. 322.20 g/mol, obtained by adding: molar mass of \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) (142.04 g/mol) and 10 times the molar mass of water (10*18.02 g/mol).

Step 2: Calculate Moles of Anhydrous Sodium Sulfate

Next, calculate the number of moles of anhydrous \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) in the given 24.05g. This can be done using the formula: moles = mass / molar mass. Plug in the values to get: moles of \(\mathrm{Na}_{2} \mathrm{SO}_{4}=\) 24.05g / 142.04 g/mol = 0.169 mol.

Step 3: Calculate Mass Increase

Using stoichiometry, the same number of moles of decahydrate will be formed from anhydrous \(\mathrm{Na}_{2} \mathrm{SO}_{4}\). Therefore, calculate the mass of decahydrate produced using the formula: mass = moles * molar mass. Substitute the values to get: mass of \(\mathrm{Na}_{2} \mathrm{SO}_{4} \cdot 10 \mathrm{H}_{2} \mathrm{O}=\) 0.169 mol * 322.20 g/mol = 54.45g. Finally, subtract the initial mass of anhydrous \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) from the final mass of the decahydrate to find the increase in mass. That is: 54.45g - 24.05g = 30.4g. So, the mass increases by 30.4g when all the anhydrous sodium sulfate is converted to the decahydrate.

Key Concepts

Anhydrous Sodium Sulfate

Anhydrous sodium sulfate, with the chemical formula \(\mathrm{Na}_{2} \mathrm{SO}_{4}\), is a compound composed of sodium, sulfur, and oxygen. It is a white, crystalline solid often used in detergents and manufacturing. To understand its role in chemical processes, it's essential to know about its molar mass and composition.
The molar mass of anhydrous sodium sulfate is calculated based on its constituent elements:

• 2 sodium atoms, \((2 \times 23.0)\ \mathrm{g/mol}\)
• 1 sulfur atom, \(32.1\ \mathrm{g/mol}\)
• 4 oxygen atoms, \((4 \times 16.0)\ \mathrm{g/mol}\)

By adding these values together, we find that the molar mass is approximately \(142.04\ \mathrm{g/mol}\). This value is crucial for converting grams to moles, which is a typical step in stoichiometric calculations.

Decahydrate Formation

When anhydrous sodium sulfate absorbs water vapor, it transforms into a decahydrate form, represented as \(\mathrm{Na}_{2} \mathrm{SO}_{4} \cdot 10 \mathrm{H}_{2} \mathrm{O}\). This means the sulfate compound incorporates ten water molecules into its crystalline structure. This transformation is vital in many industrial and chemical processes as it dramatically increases the compound's mass.
The molar mass of the decahydrate is substantially higher due to the added water molecules. As previously calculated, it totals approximately \(322.20\ \mathrm{g/mol}\):

• Molar mass of anhydrous sodium sulfate, \(142.04\ \mathrm{g/mol}\)
• Plus ten water molecules, \(10 \times 18.02\ \mathrm{g/mol}\)

Understanding the structural change during decahydrate formation is essential for predicting changes in mass and calculating reagent quantities needed in reactions.

Stoichiometry Calculation

Stoichiometry involves calculating the quantities of reactants and products in chemical reactions. When dealing with anhydrous sodium sulfate and its conversion to decahydrate, stoichiometry helps determine the mass increase.
In this situation, you start by finding the number of moles of \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) using the formula:

• Moles = Mass / Molar Mass

For \(24.05 \mathrm{g}\) of anhydrous sodium sulfate, the moles calculate as \(0.169 \ \mathrm{mol}\).
Since the reaction completely converts to the decahydrate form, the same number of moles applies to \(\mathrm{Na}_{2} \mathrm{SO}_{4} \cdot 10 \mathrm{H}_{2} \mathrm{O}\). Hence, calculate the new mass by multiplying the moles by the molar mass of the decahydrate. This results in: \[0.169\ \mathrm{mol} \times 322.20\ \mathrm{g/mol} = 54.45\ \mathrm{g}\]
Subtracting the initial mass of anhydrous sodium sulfate from the mass of the decahydrate gives the increase in mass:\[54.45\ \mathrm{g} - 24.05\ \mathrm{g} = 30.4\ \mathrm{g}\]
This stoichiometric approach allows precise prediction of product mass changes in hydration reactions.