Answer :
Certainly! Let's solve the problem step-by-step to find out the percentage by mass of aluminium in aluminium sulphate, [tex]\( \text{Al}_2(\text{SO}_4)_3 \)[/tex].
### Step 1: Determine the Atomic Masses
The atomic masses given in the question are:
- Aluminium (Al): 27 g/mol
- Sulphur (S): 32 g/mol
- Oxygen (O): 16 g/mol
### Step 2: Determine the Molecular Formula of Aluminium Sulphate
The molecular formula for aluminium sulphate is [tex]\( \text{Al}_2(\text{SO}_4)_3 \)[/tex].
### Step 3: Calculate the Molar Mass of Aluminium Sulphate
To find the molar mass of [tex]\( \text{Al}_2(\text{SO}_4)_3 \)[/tex], we need to sum the atomic masses of all atoms in the formula:
- Aluminium: There are 2 atoms of Al, so [tex]\( 2 \times 27 = 54 \)[/tex] g/mol.
- Sulphur: There are 3 atoms of S, so [tex]\( 3 \times 32 = 96 \)[/tex] g/mol.
- Oxygen: Each [tex]\( \text{SO}_4 \)[/tex] unit contains 4 atoms of O and there are 3 [tex]\( \text{SO}_4 \)[/tex] units. So, [tex]\( 4 \times 3 = 12 \)[/tex] atoms of O in total. Therefore, [tex]\( 12 \times 16 = 192 \)[/tex] g/mol.
Adding these together, we get the molar mass of [tex]\( \text{Al}_2(\text{SO}_4)_3 \)[/tex]:
[tex]\[ 54 \text{ (from Al)} + 96 \text{ (from S)} + 192 \text{ (from O)} = 342 \text{ g/mol} \][/tex]
### Step 4: Calculate the Mass Contribution of Aluminium
The mass of aluminium in one mole of [tex]\( \text{Al}_2(\text{SO}_4)_3 \)[/tex] is:
[tex]\[ 2 \times 27 \, \text{g} = 54 \, \text{g} \][/tex]
### Step 5: Calculate the Percentage by Mass of Aluminium
To find the percentage by mass of aluminium in the compound, we use the formula:
[tex]\[ \text{Percentage of Al} = \left( \frac{\text{Mass of Al}}{\text{Molar mass of } \text{Al}_2(\text{SO}_4)_3} \right) \times 100 \% \][/tex]
Substitute the values:
[tex]\[ \text{Percentage of Al} = \left( \frac{54 \, \text{g}}{342 \, \text{g/mol}} \right) \times 100 \% \approx 15.79 \% \][/tex]
Thus, the percentage by mass of aluminium in aluminium sulphate [tex]\( \text{Al}_2(\text{SO}_4)_3 \)[/tex] is approximately 15.79%.
### Step 1: Determine the Atomic Masses
The atomic masses given in the question are:
- Aluminium (Al): 27 g/mol
- Sulphur (S): 32 g/mol
- Oxygen (O): 16 g/mol
### Step 2: Determine the Molecular Formula of Aluminium Sulphate
The molecular formula for aluminium sulphate is [tex]\( \text{Al}_2(\text{SO}_4)_3 \)[/tex].
### Step 3: Calculate the Molar Mass of Aluminium Sulphate
To find the molar mass of [tex]\( \text{Al}_2(\text{SO}_4)_3 \)[/tex], we need to sum the atomic masses of all atoms in the formula:
- Aluminium: There are 2 atoms of Al, so [tex]\( 2 \times 27 = 54 \)[/tex] g/mol.
- Sulphur: There are 3 atoms of S, so [tex]\( 3 \times 32 = 96 \)[/tex] g/mol.
- Oxygen: Each [tex]\( \text{SO}_4 \)[/tex] unit contains 4 atoms of O and there are 3 [tex]\( \text{SO}_4 \)[/tex] units. So, [tex]\( 4 \times 3 = 12 \)[/tex] atoms of O in total. Therefore, [tex]\( 12 \times 16 = 192 \)[/tex] g/mol.
Adding these together, we get the molar mass of [tex]\( \text{Al}_2(\text{SO}_4)_3 \)[/tex]:
[tex]\[ 54 \text{ (from Al)} + 96 \text{ (from S)} + 192 \text{ (from O)} = 342 \text{ g/mol} \][/tex]
### Step 4: Calculate the Mass Contribution of Aluminium
The mass of aluminium in one mole of [tex]\( \text{Al}_2(\text{SO}_4)_3 \)[/tex] is:
[tex]\[ 2 \times 27 \, \text{g} = 54 \, \text{g} \][/tex]
### Step 5: Calculate the Percentage by Mass of Aluminium
To find the percentage by mass of aluminium in the compound, we use the formula:
[tex]\[ \text{Percentage of Al} = \left( \frac{\text{Mass of Al}}{\text{Molar mass of } \text{Al}_2(\text{SO}_4)_3} \right) \times 100 \% \][/tex]
Substitute the values:
[tex]\[ \text{Percentage of Al} = \left( \frac{54 \, \text{g}}{342 \, \text{g/mol}} \right) \times 100 \% \approx 15.79 \% \][/tex]
Thus, the percentage by mass of aluminium in aluminium sulphate [tex]\( \text{Al}_2(\text{SO}_4)_3 \)[/tex] is approximately 15.79%.
