Carbon monoxide emitted by automobiles is an environmental hazard . A car has an engine of four cylinders with a total cylinder volume of 1600 cc and a fuel consumption fo 7.0 `dm^(3)` per 100 Km , when driving at an average speed of 80 km/hr . In one second , each cylinder goes through 25 burn cycles and consumes 0.400g of fuel . The compression ratio, which is the ratio between the smallest and largest volume within the cylinder as the piston moves forward and backward is 1:8.
Calculate the air intake of the engine `(m^(3)S^(-1))` , if the gaseous fuel and air are introduced into the cylinder when its volume is largest until the pressure is 101.0 KPa. the temperature of both incoming air and fuel is `100^(@)`C. (Assume the fuel to be isoctane , `C_(8)H_(18))`

Answer

To calculate the air intake of the engine in \( m^3/s \), we will follow these steps:

### Step 1: Calculate the total fuel consumption per second
Given that the car consumes 0.400 g of fuel per burn cycle and each cylinder goes through 25 burn cycles per second, we can calculate the total fuel consumption per second for all four cylinders.

\[ \text{Total fuel consumption per second} = \text{Fuel per cycle} \times \text{Burn cycles per second} \times \text{Number of cylinders} \]

\[ = 0.400 \, \text{g/cycle} \times 25 \, \text{cycles/s} \times 4 \, \text{cylinders} = 40 \, \text{g/s} \]

### Step 2: Convert fuel consumption to moles
The molar mass of iso-octane \( C_8H_{18} \) is approximately 114.22 g/mol. We can convert the fuel consumption from grams to moles.

\[ \text{Moles of fuel per second} = \frac{\text{Total fuel consumption per second}}{\text{Molar mass of iso-octane}} = \frac{40 \, \text{g/s}}{114.22 \, \text{g/mol}} \approx 0.350 \, \text{mol/s} \]

### Step 3: Determine the stoichiometric air requirement
The combustion reaction for iso-octane is:

\[ C_8H_{18} + 12.5 O_2 \rightarrow 8 CO_2 + 9 H_2O \]

From the reaction, we see that 1 mole of iso-octane requires 12.5 moles of \( O_2 \). Since air is approximately 21% oxygen by volume, we can calculate the volume of air required.

\[ \text{Moles of air required} = 12.5 \, \text{moles of } O_2 \times \frac{1}{0.21} \approx 59.52 \, \text{moles of air} \]

### Step 4: Calculate the air intake in moles per second
Now we can calculate the moles of air required per second based on the moles of iso-octane consumed.

\[ \text{Moles of air per second} = \text{Moles of fuel per second} \times 59.52 \approx 0.350 \, \text{mol/s} \times 59.52 \approx 20.82 \, \text{mol/s} \]

### Step 5: Convert moles of air to volume
Using the ideal gas law, we can calculate the volume of air at the given conditions (temperature and pressure). We will use the ideal gas equation:

\[ PV = nRT \]

Where:
- \( P = 101.0 \, \text{kPa} = 101000 \, \text{Pa} \)
- \( n = 20.82 \, \text{mol/s} \)
- \( R = 8.314 \, \text{J/(mol K)} \)
- \( T = 100 \, \text{°C} = 373.15 \, \text{K} \)

Rearranging for \( V \):

\[ V = \frac{nRT}{P} \]

Substituting the values:

\[ V = \frac{20.82 \, \text{mol/s} \times 8.314 \, \text{J/(mol K)} \times 373.15 \, \text{K}}{101000 \, \text{Pa}} \approx 0.052 \, m^3/s \]

### Final Answer
The air intake of the engine is approximately \( 0.052 \, m^3/s \).

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