The internal engine of a 1200 -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: See the Chemistry in Action essay "The Efficiency of Heat Engines" in Section \(17.5 .\) 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.]

Understanding the Carnot efficiency is essential when it comes to heat engines and their capabilities. In simple terms, the Carnot efficiency represents the idealized efficiency of a heat engine that operates on the Carnot cycle. This efficiency is determined by the temperatures at which the engine operates, specifically the temperature of the heat source (\texttt{T_h}) and the temperature of the heat sink (\texttt{T_c}).

The Carnot efficiency is calculated using the formula: \( \eta = 1 - \frac{T_c}{T_h} \) where \( \eta \) is the efficiency, \( T_c \) is the absolute temperature of the cold reservoir, and \( T_h \) is the absolute temperature of the hot reservoir, both measured in Kelvin.

Now, why is Carnot efficiency important? It sets the maximum possible efficiency that any heat engine working between two temperatures could achieve. No real engine can be more efficient than a Carnot engine due to inherent losses and irreversible processes. In our exercise, the Carnot efficiency provided us a theoretical benchmark to calculate the maximum work output possible from the fuel consumed by the car's engine.

In chemistry and physics, the molar mass is a fundamental concept that connects the mass of a substance to the number of particles or moles of that substance. It is especially crucial when dealing with chemical reactions and conversions between mass and moles of a substance.

Molar mass is defined as the mass of one mole of a substance and is usually expressed in grams per mole (\texttt{g/mol}). The numerical value of the molar mass is equal to the atomic or molecular weight of the substance taken from the periodic table. For a compound like octane (\texttt{C8H18}), you would calculate its molar mass by summing the molar masses of all the atoms in the molecule: \( 8 \times 12.01 \texttt{ g/mol} \texttt{(carbon)} + 18 \times 1.008 \texttt{ g/mol} \texttt{(hydrogen)} = 114.22 \texttt{ g/mol} \) for octane.

Knowing the molar mass allowed us to convert the mass of octane fuel into moles in the exercise solution. This step was extremely important to then determine the total heat energy that could be produced by the fuel's combustion.

Heat engines are devices that convert heat energy into mechanical work. They play a pivotal role in many aspects of modern society, from powering vehicles to generating electricity. The basic operation of a heat engine involves taking in heat energy from a high-temperature source, doing work, and expelling some waste heat to a lower temperature sink.

The efficiency of a heat engine is a measure of how well it can convert the input heat into useful work. As per the second law of thermodynamics, no heat engine can be 100% efficient due to the inevitable waste heat that must be released.

In our exercise, we utilized the concept of heat engines to determine the maximum height a car could climb utilizing the energy obtained from the combustion of octane. Neglecting friction and other losses, the efficiency of this conversion from chemical energy to gravitational potential energy can be idealized through Carnot's principle. However, one must remember that real-world engines have various inefficiencies and usually operate at efficiencies much lower than the theoretical Carnot efficiency.