In a rocket motor fuelled with Heptane `(C_(7)H_(16))` How many kg of liquid oxygen should be provided with each kg of heptane sample (which is `50%` pure) for its complete combustion :-

Options

A. `0.88 kg`
B. `0.52 kg`
C. `1.76kg`
D. `2.64 kg`

Answer

To solve the problem of determining how much liquid oxygen is needed for the complete combustion of heptane (C₇H₁₆) in a rocket motor, we will follow these steps:

### Step 1: Write the balanced chemical equation for the combustion of heptane.
The balanced equation for the complete combustion of heptane is:
\[ C_7H_{16} + O_2 \rightarrow CO_2 + H_2O \]

### Step 2: Balance the equation.
To balance the combustion reaction:
1. For carbon: 7 carbon atoms in heptane produce 7 CO₂.
2. For hydrogen: 16 hydrogen atoms in heptane produce 8 H₂O.
3. Now, we need to balance the oxygen atoms.
- The total number of oxygen atoms needed = (7 CO₂ × 2) + (8 H₂O × 1) = 14 + 8 = 22 oxygen atoms.
- Since O₂ has 2 oxygen atoms, we need 22/2 = 11 O₂ molecules.

Thus, the balanced equation is:
\[ C_7H_{16} + 11 O_2 \rightarrow 7 CO_2 + 8 H_2O \]

### Step 3: Calculate the molar mass of heptane and oxygen.
- Molar mass of heptane (C₇H₁₆):
\[ 7 \times 12 + 16 \times 1 = 84 + 16 = 100 \text{ g/mol} \]

- Molar mass of oxygen (O₂):
\[ 2 \times 16 = 32 \text{ g/mol} \]

### Step 4: Determine the amount of oxygen required for 1 kg of heptane.
From the balanced equation, 1 mole (100 g) of heptane reacts with 11 moles of O₂:
- Therefore, 100 g of heptane requires:
\[ 11 \text{ moles of } O_2 \times 32 \text{ g/mol} = 352 \text{ g of } O_2 \]

### Step 5: Calculate the amount of oxygen required for 1 kg of heptane.
Since 1 kg = 1000 g, we can calculate the oxygen needed for 1 kg of heptane:
- For 1 kg of heptane:
\[ \frac{352 \text{ g O}_2}{100 \text{ g C}_7H_{16}} \times 1000 \text{ g C}_7H_{16} = 3520 \text{ g O}_2 \]
\[ = 3.52 \text{ kg O}_2 \]

### Step 6: Adjust for the purity of heptane.
Given that the heptane sample is only 50% pure, the effective amount of heptane is:
- Actual heptane in 1 kg sample:
\[ 50\% \text{ of } 1 \text{ kg} = 0.5 \text{ kg} \]

### Step 7: Calculate the amount of oxygen needed for 0.5 kg of heptane.
Now, we need to find the amount of oxygen required for 0.5 kg of heptane:
- If 3.52 kg of O₂ is needed for 1 kg of heptane, then for 0.5 kg of heptane:
\[ 3.52 \text{ kg O}_2 \times 0.5 = 1.76 \text{ kg O}_2 \]

### Conclusion:
Thus, the amount of liquid oxygen required for the complete combustion of 1 kg of the 50% pure heptane sample is **1.76 kg**.

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