To find the relative molecular mass of [tex]\( X \)[/tex] in the mixture described, we can follow a structured step-by-step approach:
### Step 1: Determine the mass contribution of [tex]\( Y \)[/tex]
1. Find the number of molecules of [tex]\( Y \)[/tex]:
[tex]\[ \text{Number of molecules of } Y = 1.85 \times 10^{21} \][/tex]
2. Given the relative molecular mass of [tex]\( Y \)[/tex] is 42.0 (atomic mass units).
3. Use Avogadro's number, [tex]\( 6.022 \times 10^{23} \)[/tex], to convert the number of molecules to moles:
[tex]\[
\text{Moles of } Y = \frac{\text{Number of molecules of } Y}{6.022 \times 10^{23}}
= \frac{1.85 \times 10^{21}}{6.022 \times 10^{23}}
\][/tex]
4. Calculate the mass of [tex]\( Y \)[/tex] in grams:
[tex]\[
\text{Mass of } Y = \left( \frac{1.85 \times 10^{21}}{6.022 \times 10^{23}} \right) \times 42.0
= 0.12902690136167388 \, \text{g}
\][/tex]
### Step 2: Find the mass of [tex]\( X \)[/tex]
1. The total mass of the mixture is 0.688 grams.
2. Subtract the mass of [tex]\( Y \)[/tex] from the total mass to find the mass of [tex]\( X \)[/tex]:
[tex]\[
\text{Mass of } X = 0.688 \, \text{g} - 0.12902690136167388 \, \text{g}
= 0.5589730986383261 \, \text{g}
\][/tex]
### Step 3: Determine the relative molecular mass of [tex]\( X \)[/tex]
1. Find the number of molecules of [tex]\( X \)[/tex]:
[tex]\[ \text{Number of molecules of } X = 1.65 \times 10^{21} \][/tex]
2. Convert the number of molecules of [tex]\( X \)[/tex] to moles:
[tex]\[
\text{Moles of } X = \frac{1.65 \times 10^{21}}{6.022 \times 10^{23}}
\][/tex]
3. Calculate the relative molecular mass of [tex]\( X \)[/tex] using the mass of [tex]\( X \)[/tex] and the number of moles calculated:
[tex]\[
\text{Relative molecular mass of } X = \left( \frac{0.5589730986383261 \, \text{g} \times 6.022 \times 10^{23}}{1.65 \times 10^{21}} \right)
= 204.0082424242424
\][/tex]
### Conclusion
The relative molecular mass of [tex]\( X \)[/tex] is approximately 204.008.
