A particular car does work at the rate of about 7.0 kJ/s...

A particular car does work at the rate of about 7.0 kJ/s when traveling at a steady 21.8 m/s along a level road. This is the work done against friction. The car can travel 17 km on 1.0 L of gasoline at this speed (about ). What is the minimum value for $T_H$ if $T_L$ is 25$^\circ$C? The energy available from 1.0 L of gas is $3.2 \times 10^7$ J.

Key Concepts

Carnot Efficiency

The Carnot efficiency is the maximum possible efficiency that any heat engine operating between two thermal reservoirs can achieve. It is defined by the equation ? = 1 - (T_L/T_H), where T_L is the temperature of the low?temperature reservoir and T_H is the temperature of the high?temperature reservoir, expressed in absolute units (Kelvin). This concept sets an upper limit on how effectively chemical or thermal energy can be converted into work by any engine, and it is central to understanding the limitations imposed by the laws of thermodynamics.

Energy Conversion and Efficiency

Energy conversion involves transforming energy from one form (in this case, chemical energy in gasoline) to another form, like mechanical work. The efficiency of this conversion process is the ratio of the useful work output to the total energy input. Being able to relate the work required to overcome forces (such as friction) to the energy provided by fuel helps determine the minimum performance requirements of an engine, and it illustrates how inefficiencies and losses dictate engine design and operating temperatures.

Work, Power, and Energy

Work is the energy transferred when a force is applied over a distance, while power is the rate at which work is done. In practical applications, calculating the work done against resistive forces over a given distance or time, using the power output and operating speed, is essential to determine how much energy is needed. Understanding the relationship between these quantities aids in designing systems that must meet specific energy and performance criteria.

Temperature Scales in Thermodynamics

In thermodynamics, temperature must be measured in absolute units (Kelvin) when used in formulas such as the Carnot efficiency equation. Converting temperatures from degrees Celsius to Kelvin is a necessary step for consistency and accuracy in calculations involving ratios or differences of temperatures, since the Kelvin scale starts at absolute zero, making it the appropriate scale for these analyses.

Transcript

00:01 So the minimum value for th would occur if the engine were a carnot engine.

00:09 So we calculate the efficiency of a carnot engine from the given data and then use that to calculate th.

00:20 So the power output, which is the power to move the car, is equal to wart turn over time.

00:31 And the power input which is the power to power from the gasoline will be equal to the heat input over time and this is equal to 7 ,000 watt and this will be equal to so we need to calculate this so we have 10 to the power 3 .2 times 10 to the power 7 jules in one liter so that much of gasoline in one liter.

01:09 And we need to express this power, to calculate this power in units of watts, so that we have the same units for both the powers.

01:19 And for that, we need to do a little bit of conversion.

01:22 So first we convert the liter into meters.

01:26 So one liter has 17 ,000 meter.

01:35 So basically that came from the information provided in the problem that the car can travel 1 ,100 kilometers on 1 liter of gasoline.

01:50 So that's how we get this conversion.

01:53 So in 1 liter, it travels 17 ,000 meter.

01:57 So that way we have this unit in jules per meter.

02:04 So how much gasoline is expended.

02:07 In one meter or how much gasoline we need in one meter now then we need to convert this meter into seconds so for that we have the velocity so it's saying that the car travels at a steady velocity or steady speed of 21 .8 meter per second so we can simply multiply that factor so, 21 .8 meter per second, which is the velocity...

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