The internal combustion engine of a \(1200-\mathrm{kg}\) car is designed to run on octane \(\left(\mathrm{C}_{8} \mathrm{H}_{18}\right),\) whose enthalpy of combustion is \(5510 \mathrm{~kJ} / \mathrm{mol}\). If the car is moving up a slope, calculate the maximum height (in meters) to which the car can be driven on 1.0 gallon of the fuel. Assume that the engine cylinder temperature is \(2200^{\circ} \mathrm{C}\) and the exit temperature is \(760^{\circ} \mathrm{C},\) and neglect all forms of friction. The mass of 1 gallon of fuel is \(3.1 \mathrm{~kg} .\) [Hint: The efficiency of the internal combustion engine, defined as work performed by the engine divided by the energy input, is given by \(\left(T_{2}-T_{1}\right) / T_{2},\) where \(T_{2}\) and \(T_{1}\) are the engine's operating temperature and exit temperature (in kelvins). The work done in moving the car over a vertical distance is \(m g h,\) where \(m\) is the mass of the car in \(\mathrm{kg}, g\) the acceleration due to gravity \(\left(9.81 \mathrm{~m} / \mathrm{s}^{2}\right),\) and \(h\) the height in meters. \(]\)

Step by step solution

01

Calculate the Energy Available from Fuel

Knowing that 1 mol of octane (\(\mathrm{C}_{8} \mathrm{H}_{18}\)) releases 5510 kJ of energy, we'll determine the number of moles in 3.1 kg of octane, which is the weight of 1 gallon of fuel. The molar Mass of octane is 114 g/mol. So, first convert the mass of fuel from kg to g, then compute the number of moles (n) with formula \(n=m/\text{Molar Mass}\). Then, calculate the total energy (E) released with the formula \(E=n \times \text{Enthalpy}\).

02

Compute the Efficiency of the Engine

The efficiency (\(\eta\)) of the engine is given by \(\eta=\frac{(T_2 - T_1)}{T_2}\), where \(T_1\) and \(T_2\) are the temperatures of the engine's cylinder and exhaust, respectively, in kelvins. Note that convert the temperatures from Celsius to Kelvin by adding 273 to each. Then, calculate \(\eta\).

03

Calculate the Work Done by the Engine

The work (W) done by engine is given by \(W=\eta \times E\). Calculate the actual work performed by the engine using this formula.

04

Calculate the Maximum Height

The work done by engine is equal to the potential energy gained by the car, which is given as \(W=mgh\), where m is mass of the car, g is acceleration due to gravity, and h is the height to which the car can rise. Here, \(W\), \(m\), and \(g\) are known so calculate height \(h\) using the rearranged formula \(h=\frac{W}{mg}\).

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