The hydrocarbon (X) with molar mass 80 g mol\(^{-1}\) and 90%carbon has ________________ degree of unsaturation.
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Approach Solution - 1
Step 1: Determine the Formula of the Hydrocarbon
We are given that the molar mass of the hydrocarbon (X) is 80 g/mol and it contains 90% carbon by mass. To find the degree of unsaturation, we first need to determine the empirical formula of the compound.
The mass of carbon in 80 g of the compound is 90% of 80, which is:
Mass of carbon = \( 0.90 \times 80 = 72 \, \text{g} \)
The mass of hydrogen is the remainder, which is:
Mass of hydrogen = \( 80 - 72 = 8 \, \text{g} \)
Step 2: Calculate Moles of Carbon and Hydrogen
Now, calculate the moles of carbon and hydrogen using their respective atomic masses (C = 12 g/mol and H = 1 g/mol):
Moles of carbon = \( \frac{72 \, \text{g}}{12 \, \text{g/mol}} = 6 \, \text{mol} \)
Moles of hydrogen = \( \frac{8 \, \text{g}}{1 \, \text{g/mol}} = 8 \, \text{mol} \)
Step 3: Determine the Empirical Formula
The mole ratio of carbon to hydrogen is 6:8, which simplifies to 3:4. Therefore, the empirical formula of the compound is C3H4.
Step 4: Calculate the Molecular Formula
The molar mass of the empirical formula C3H4 is:
Molar mass of C3H4 = \( 3 \times 12 + 4 \times 1 = 36 + 4 = 40 \, \text{g/mol} \)
The given molar mass of the compound is 80 g/mol. To find the molecular formula, divide the molar mass of the compound by the molar mass of the empirical formula:
Ratio = \( \frac{80}{40} = 2 \)
This means the molecular formula is twice the empirical formula, or C6H8.
Step 5: Calculate the Degree of Unsaturation
The degree of unsaturation (DU) can be calculated using the formula:
\[ \text{DU} = \frac{2C + 2 - H}{2} \]
For C6H8, we have:
DU = \( \frac{2(6) + 2 - 8}{2} = \frac{12 + 2 - 8}{2} = \frac{6}{2} = 3 \)
Conclusion
The hydrocarbon (X) with a molar mass of 80 g/mol and 90% carbon has a degree of unsaturation of 3.
Approach Solution -2
The degree of unsaturation (DU) indicates the number of rings and multiple bonds (double or triple bonds) in a compound. The formula for calculating the degree of unsaturation is given by:
\[ \text{DU} = \frac{2C + 2 - H}{2}, \] where \( C \) is the number of carbon atoms and \( H \) is the number of hydrogen atoms in the compound.
Step 2: Use the given information.
We are given the molar mass of the hydrocarbon \( X \) as 80 g/mol and it contains 90% carbon. From this, we can calculate the number of carbon and hydrogen atoms.
- Molar mass of \( X \) = 80 g/mol
- Mass of carbon in \( X \) = 90% of 80 g = 72 g
- Moles of carbon (\( C \)) = \( \frac{72 \, \text{g}}{12 \, \text{g/mol}} = 6 \, \text{mol} \)
- Mass of hydrogen in \( X \) = 10% of 80 g = 8 g
- Moles of hydrogen (\( H \)) = \( \frac{8 \, \text{g}}{1 \, \text{g/mol}} = 8 \, \text{mol} \)
Thus, the hydrocarbon has 6 carbon atoms and 8 hydrogen atoms.
Step 3: Calculate the degree of unsaturation.
Using the formula for degree of unsaturation:
\[ \text{DU} = \frac{2(6) + 2 - 8}{2} = \frac{12 + 2 - 8}{2} = \frac{6}{2} = 3. \]
Final Answer:
\[ \boxed{3}. \]
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