One type of artificial diamond (commonly called YAG for yttrium aluminium garnet) can be represented by the formula `Y_(3)Al_(5)O_(12)`
`(a)` Calculate the weight percentage composition of this compound.
`(b)` What is the weight of yttrium present in a `200 -` carat `YAG` if `1` carat `-200mg` ? `(Y=89),Al=27)`

Explanation

To solve the problem step by step, we will break it down into two parts as given in the question.

### Part (a): Calculate the weight percentage composition of YAG (Y₃Al₅O₁₂)

1. **Identify the molar masses of the elements:**
- Yttrium (Y) = 89 g/mol
- Aluminum (Al) = 27 g/mol
- Oxygen (O) = 16 g/mol

2. **Calculate the total molar mass of YAG (Y₃Al₅O₁₂):**
- For Yttrium: \( 3 \times 89 = 267 \, \text{g/mol} \)
- For Aluminum: \( 5 \times 27 = 135 \, \text{g/mol} \)
- For Oxygen: \( 12 \times 16 = 192 \, \text{g/mol} \)

Now, sum these values to find the total molar mass:
\[ \text{Total molar mass} = 267 + 135 + 192 = 594 \, \text{g/mol} \]

3. **Calculate the weight percentage of each element:**
- **Weight percentage of Yttrium (Y):**
\[ \text{Weight \% of Y} = \left( \frac{267}{594} \right) \times 100 \approx 44.95\% \]

- **Weight percentage of Aluminum (Al):**
\[ \text{Weight \% of Al} = \left( \frac{135}{594} \right) \times 100 \approx 22.73\% \]

- **Weight percentage of Oxygen (O):**
\[ \text{Weight \% of O} = 100 - (44.95 + 22.73) \approx 32.32\% \]

4. **Final weight percentage composition:**
- Yttrium (Y): 44.95%
- Aluminum (Al): 22.73%
- Oxygen (O): 32.32%

### Part (b): Calculate the weight of yttrium in 200 carats of YAG

1. **Convert carats to grams:**
- Given that 1 carat = 200 mg, we convert 200 carats to grams:
\[ 200 \, \text{carats} = 200 \times 200 \, \text{mg} = 40000 \, \text{mg} = 40 \, \text{g} \]

2. **Calculate the weight of yttrium in 40 grams of YAG:**
- We know that Yttrium constitutes approximately 44.95% of YAG.
\[ \text{Weight of Y} = \left( \frac{44.95}{100} \right) \times 40 \, \text{g} \approx 17.98 \, \text{g} \]

### Summary of Results:
- **Weight percentage composition:**
- Yttrium (Y): 44.95%
- Aluminum (Al): 22.73%
- Oxygen (O): 32.32%

- **Weight of yttrium in 200 carats of YAG:** 17.98 g