A car's internal combustion engine can be modeled as a heat engine operating between a combustion temperature of 1500∘C and an air temperature of 20∘C with 30% of the Carnot efficiency. The heat of combustion of gasoline is 47 kJ/g.
Q) What mass of gasoline is burned to accelerate a 1500 kg car from rest to a speed of 20 m/s?
I got the work net value with kinetic energy formula, 300kJ, and 30% of the carnet efficiency, so I got Q_h which is 1000kJ. When to try to get the mass of gasoline burned using the info, heat of combustion of gasoline 47kHJ/g, which I got 21.28g, but it is wrong. Can anyone explain this for me?
To find the mass of gasoline burned, we need to calculate the work done by the car's engine and then convert it to energy required in terms of gasoline.
First, let's calculate the work done by the engine to accelerate the car. The work done is given by the formula:
W = (1/2)mv^2
Where:
W is the work done
m is the mass of the car
v is the final velocity
Substituting the given values:
m = 1500 kg
v = 20 m/s
W = (1/2)(1500 kg)(20 m/s)^2
W = (1/2)(1500 kg)(400 m^2/s^2)
W = 300,000 J
Next, we know that the internal combustion engine of a car is a heat engine operating between a combustion temperature of 1500°C and an air temperature of 20°C, with 30% of the Carnot efficiency.
The Carnot efficiency of a heat engine is given by the formula:
η = 1 - (Tc/Th)
Where:
η is the Carnot efficiency
Tc is the absolute temperature at which heat is rejected (in Kelvin)
Th is the absolute temperature at which heat is absorbed (in Kelvin)
Substituting the given values:
η = 1 - (20 + 273)/(1500 + 273)
η = 1 - 293/1773
η ≈ 0.834
We are given that the Carnot efficiency of the engine is 30%, which is 0.3 in decimal form. So we can set up the equation:
η = 0.3
0.834 = 0.3
Now we can calculate the heat generated by the engine using the work done and the Carnot efficiency. The equation for heat is given by:
Q = W / η
Substituting the values:
Q = 300,000 J / 0.834
Q = 359,712 J
Finally, we need to calculate the mass of gasoline burned. We are given that the heat of combustion of gasoline is 47 kJ/g, which is equivalent to 47,000 J/g.
So we can set up the equation:
Q = mc ΔH
Where:
Q is the heat generated by the engine
m is the mass of gasoline burned
c is the specific heat capacity of gasoline
ΔH is the heat of combustion of gasoline (in J/g)
Substituting the values:
359,712 J = m (47,000 J/g)
Solving for m:
m = 359,712 J / 47,000 J/g
m = 7.65 g
Therefore, approximately 7.65 grams of gasoline would be burned to accelerate the 1500 kg car from rest to a speed of 20 m/s.
To find the mass of gasoline burned, we need to calculate the work done by the car's engine to accelerate the car, and then use the given information about the car's efficiency and heat of combustion to determine the mass of gasoline required.
Step 1: Calculate the work done by the engine to accelerate the car.
The work done to accelerate the car can be calculated using the work-energy principle. The change in kinetic energy is equal to the work done.
Change in kinetic energy = (1/2) * mass * (final velocity^2 - initial velocity^2)
Given:
Mass of the car (m) = 1500 kg
Initial velocity (v_initial) = 0 m/s
Final velocity (v_final) = 20 m/s
Change in kinetic energy = (1/2) * 1500 kg * (20 m/s)^2
Step 2: Calculate the energy input to the engine.
Since the engine is an idealized Carnot engine, the efficiency (η) is given as 30% or 0.30. The efficiency of a heat engine is defined as the ratio of useful work output to the energy input. In this case, the useful work output is the change in kinetic energy calculated in step 1.
Efficiency (η) = useful work output / energy input
Energy input = useful work output / efficiency
Energy input = (Change in kinetic energy) / 0.30
Step 3: Calculate the heat generated by the combustion of gasoline.
For every gram of gasoline burned, there is a known amount of heat released, which is given as 47 kJ/g.
Step 4: Calculate the mass of gasoline burned.
The mass of gasoline burned can be calculated by dividing the energy input to the engine (from step 2) by the heat of combustion of gasoline (from step 3).
Mass of gasoline burned = Energy input / Heat of combustion of gasoline
Now, you can substitute the values into the equations and calculate the mass of gasoline burned.