A car's internal combustion engine can be modeled as a heat engine operating between a combustion temperature of 1500℃ and an air temperature of 20℃ with 30% of the Carnot efficiency. The heat of combustion of gasoline is 47 kJ/g. What mass of gasoline is burned to accelerate a 1500 kg car from rest to a speed of 30 m/s?

Verified step by step guidance

1. Step 1: Convert the temperatures from Celsius to Kelvin. The Carnot efficiency depends on the absolute temperatures, so add 273.15 to each temperature. The combustion temperature becomes 1500 + 273.15 = 1773.15 K, and the air temperature becomes 20 + 273.15 = 293.15 K.
2. Step 2: Calculate the Carnot efficiency using the formula: $\eta =(T_h-T_c)/T_h$, where Th is the combustion temperature and Tc is the air temperature. Substitute the values to find the Carnot efficiency.
3. Step 3: Multiply the Carnot efficiency by 30% to find the actual efficiency of the engine. Use the formula: $\eta =\eta C_{\mathrm{arnot}}\times 0.3$, where ηCarnot is the Carnot efficiency.
4. Step 4: Calculate the kinetic energy of the car using the formula: $K=\frac{1}{2}m{v}^2$, where m is the mass of the car (1500 kg) and v is the final speed (30 m/s). This gives the energy required to accelerate the car.
5. Step 5: Relate the energy required to the mass of gasoline burned. Use the formula: $m=\frac{K}{(}$, where K is the kinetic energy, η is the engine efficiency, and Q is the heat of combustion of gasoline (47 kJ/g). Rearrange and substitute the values to find the mass of gasoline burned.

Key Concepts

Carnot Efficiency

Carnot efficiency is the maximum theoretical efficiency of a heat engine operating between two temperatures, defined by the formula η = 1 - (T_cold/T_hot), where temperatures are in Kelvin. In this scenario, the engine operates at 30% of this efficiency, meaning it converts only a fraction of the heat energy from combustion into useful work, with the rest lost as waste heat.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion, calculated using the formula KE = 0.5 * m * v², where m is mass and v is velocity. For the car accelerating from rest to a speed of 30 m/s, the kinetic energy gained is crucial for determining how much energy must be supplied by burning gasoline.

Heat of Combustion

The heat of combustion is the amount of energy released when a substance, such as gasoline, is burned. It is typically expressed in kJ/g and indicates how much energy can be harnessed from a specific mass of fuel. In this problem, the heat of combustion of gasoline is 47 kJ/g, which will be used to calculate the mass of gasoline needed to provide the energy required for the car's acceleration.