Problem 32 - College Physics

Question

Entropy change due to driving. Premium gasoline pro- duces \(1.23 \times 10^{8} \mathrm{J}\) of heat per gallon when it is burned at a temperature of approximately \(400^{\circ} \mathrm{C}\) (although the amount can vary with the fuel mixture). If the car's engine is 25\(\%\) efficient, three- fourths of that heat is expelled into the air, typically at \(20^{\circ} \mathrm{C}\) . If your car gets 35 miles per gallon of gas, by how much does the car's engine change the entropy of the world when you drive 1.0 mile? Does it decrease or increase it?

Step-by-Step Solution

Answer

The entropy of the world increases by about 350 J/K per mile driven.

Step 1: Understand the Given Data

We have the following information: Heat produced per gallon of gasoline, \(Q_1 = 1.23 \times 10^8 \text{ J}\), engine efficiency, \(\eta = 0.25\), temperature of heat expulsion \(T_2 = 20^{\circ} \text{C} = 293 \text{ K}\). The car gets 35 miles/gallon, meaning for 1 mile, it uses \(\frac{1}{35}\) gallons.

Step 2: Calculate Heat Expelled per Mile

Since the engine is 25\(\%\) efficient, 75\(\%\) of the heat is expelled, which gives \(Q_2 = 0.75 \times Q_1\). First, calculate the heat produced per mile, then apply the 75\(\%\) expulsion: \[Q = \frac{1.23 \times 10^8}{35} \times 0.75 \] J.

Step 3: Calculate Entropy Change

The change in entropy \(\Delta S\) when heat \(Q\) is expelled into the surroundings at a temperature \(T_2\) is given by \(\Delta S = \frac{Q}{T_2}\). Use the calculated \(Q\) from Step 2 and \(T_2 = 293 \text{ K}\) to find \(\Delta S\): \[\Delta S = \frac{\left(\frac{1.23 \times 10^8}{35} \right) \times 0.75}{293}\] J/K.

Step 4: Determine if Entropy Increases or Decreases

Since entropy is being added to the surroundings (which is the expelled heat), and entropy generally increases with heat transfer to a cooler surroundings, \(\Delta S\) is positive, indicating an increase in the entropy of the world.

Key Concepts

Engine Efficiency

Engine efficiency is a measure of how well an engine can transform fuel energy into useful mechanical work. It’s expressed as a percentage. In the example provided, the engine efficiency is 25\(\%\). This means that only 25\(\%\) of the total energy produced by burning gasoline is converted into work to move the car.
There are several factors that affect engine efficiency:

• Quality of gasoline
• Temperature and pressure conditions
• The specific design and condition of the engine

The rest of the energy, which is 75\(\%\) in this case, is lost primarily as heat. Improving engine efficiency typically involves reducing these energy losses, ensuring that more of the fuel's energy is used for work and less is wasted. More efficient engines are better for both the environment and wallet since they require less fuel to achieve the same amount of work.

Thermodynamics

Thermodynamics deals with the principles governing the energy and work of a system. It is a fundamental concept that applies to many scientific fields. The example focuses on the car engine, a practical application of thermodynamics. Key aspects include how energy is converted from one form to another and how it transfers between systems. The first law of thermodynamics states that energy cannot be created or destroyed, only transformed. In the case of an engine, the chemical energy in fuel is transformed into mechanical energy to move the car, with some lost as heat.
The second law provides insight into the direction of these transformations. It tells us that systems naturally evolve towards states of greater disorder, or entropy. This is seen in the car engine, where the burning of fuel increases the entropy of the surroundings as most energy is expelled as waste heat. Understanding these concepts can help in designing more efficient engines and predicting how systems will behave.

Heat Transfer

Heat transfer involves the movement of heat energy from one place to another. In engines, managing heat transfer is crucial because it influences efficiency and performance. There are mainly three modes of heat transfer: conduction, convection, and radiation. In the context of the car engine:

• Conduction occurs through engine parts when heat moves directly through solid materials.
• Convection involves heat movement via fluids or gases, such as engine coolant or air around the engine.
• Radiation can also transfer heat away from the engine surface into the air.

In the given problem, 75\(\%\) of the heat is expelled into the surroundings, effectively transferring waste heat to the environment. Proper management and reduction of unwanted heat transfer help in retaining more usable energy within the system, enhancing the engine's efficiency.

Calorimetry

Calorimetry is the science of measuring heat changes in physical and chemical processes. It's essential for understanding how much energy is released or absorbed during these processes, such as fuel combustion. When evaluating an engine's performance, calorimetry helps we calculate how much heat is produced from burning gasoline and how much is converted into work versus lost as waste. From the exercise, we know

• Heat produced per gallon: \(1.23 \times 10^8\) J.
• Engine efficiency determines the fraction converted to work.
• The remaining heat is calculated and used to find out how it affects entropy and overall environmental impact.

Thus, calorimetry not only aids in quantifying energy output but also highlights the efficiency and sustainability of different engines, helping engineers to improve fuel efficiency and reduce emissions.