Advanced Placement Physics B Chapter 12: Heat Engines and Efficiency

Gas Cycles When a gas undergoes a complete cycle, it starts and ends in the same state. The gas is identical before and after the cycle, so there is no overall change in the pressure, volume, or temperature of the gas. ∆U = 0 for a complete cycle. The environment, however, has been changed.

Work done by a gas: cycle The work done by a gas is the area enclosed by the cycle’s path. If the cycle is navigated clockwise, the work done is positive; if counterclockwise, the work done is negative. The work done by the environment/surroundings will have the OPPOSITE sign.

Work done by a gas in a cyclic process Work done by gas = Area

Second Law of Thermodynamics Three forms: 1. Kelvin-Planck statement: It is impossible to extract an amount of heat Q H from a hot reservoir and use it all to do work W. Some amount of heat Q C must be exhausted to a cold reservoir. 2. Clausius statement: It is not possible for heat to flow from a colder body to a warmer body without any work having been done to accomplish this flow. Energy will not flow spontaneously from a low temperature object to a higher temperature object. 3. In any cyclic process the entropy will either increase or remain the same.

Second Law: Heat Engines Through a cyclic process, heat engines convert heat from the environment into useful work. Heat engines do not deliver heat; they transfer heat from a hot reservoir to a cold reservoir, and in the process do work. However, according to the second law the process is not totally efficient. Some heat is “dumped” into the environment (“cold reservoir”) by the engine and is not converted to work. The efficiency of an engine tells us how much heat is needed to produce a specific amount of work.

Heat engines flow diagram Q H = W + Q C Q H : Heat that is put into the system and comes from the hot reservoir in the environment. W: Work that is done by the system on the environment. Q C : Waste heat that is dumped into the cold reservoir in the environment.

Efficiency The efficiency of an engine is related what fraction of the energy put into a system is converted to useful work. In the case of a heat engine, the energy that is put in is the heat that flows into the system from the hot reservoir. Only some of the heat that flows in is converted to work. The rest is waste heat that is dumped into the cold reservoir.

Efficiency Equation Efficiency = e = W/Q H = (Q H - Q C )/Q H W: Work done by engine on environment Q H : Heat absorbed from hot reservoir Q C : Waste heat dumped to cold reservoir Efficiency is often given as percent efficiency.

Rates of Energy Use Sometimes you will be asked to find the rate at which energy is used by an engine. Rate at which heat is absorbed = Q H /t Rate at which heat is expelled = Q C /t Rate at which work is done: P = W/t

Efficiency in terms of rates e = W/Q H = (W/t)/(Q H /t) e = P/(Q H /t) P = Q H /t - Q C /t

Maximum Efficiency: Carnot Cycle In the early 1800, French engineer Sadi Carnot developed a theoretical engine cycle providing maximum efficiency. The cycle is composed of fully reversible processes: 1. An isothermal expansion at the hot temperature 2. Adiabatic expansion 3. Reversible isothermal compression at the cold temperature 4. Adiabatic compression

Carnot cycle

Carnot Efficiency Equation For a Carnot engine, the efficiency can be calculated from the temperatures of the hot and cold reservoirs. Carnot Efficiency = (T H - T C )/T H T H : Temperature of hot reservoir (K) T C : Temperature of cold reservoir (K)

Problem The efficiency of a Carnot engine is 30%. The engine absorbs 800 J of heat per cycle from a hot temperature reservoir at 500 K. Determine (a) the heat expelled per cycle and (b) the temperature of the cold reservoir

Carnot Theorem Because gas processes in reality are irreversible, the Carnot efficiency is a theoretical maximum. Carnot’s Theorem: No engine operating between two heat reservoirs can be more efficient than a Carnot engine operating between those same reservoirs. A corollary to this statement is that any two engines operating at the same hot and cold temperatures will have the same efficiency.

Entropy Entropy is often thought of as randomness, or disorder Isolated systems spontaneously move from a state of low entropy to a state of high entropy The entropy of an isolated system can not decrease This has to do with statistical probability – a state of disorder or randomness is more statistically probable Unfortunately “randomness” and “disorder” are qualitative rather than quantitative terms, so ….

We can quantify entropy Actually, what we can quantify is the change in entropy of a system, symbolized by ∆S: ∆S = Q/T Where Q = amount of heat absorbed by the system at a given temperature T

More on entropy Many scientists/textbooks do not like using the terms “disorder” and “randomness” to describe entropy and instead favor words like “multiplicity” and use the following definition: “a measure of the dispersal/spreading of energy in a chemical (or energy-related physical) change” A state which is more dispersed in space and/or energy has a higher entropy. Example: 1 mole of water vapor has more entropy than 1 mole of ice, as the water vapor is dispersed in space.

Entropy and cosmology Heat death of the universe “Time’s arrow”?