Lecture 2: The First Law of Thermodynamics

1. Lecture 2: The First Law of Thermodynamics Schroeder Ch. 4 Gould and Tobochnik: Ch. 2.8 – 2.11

2. Outline • Internal Energy • Adiabatic Processes • Carnot Cycle • The Otto Cycle • The Diesel Cycle • The Brayton Cycle

3. Introduction • The first law of thermodynamics is essentially a law of conservation of energy for macroscopic systems. • Conservation can be imposed by requiring that the variation of the total energy of the system and its environment is identically zero. • However, in order to make this precise, one should be able to say what exactly the energy of a macroscopic system is in the first place.

4. Internal Energy • The internal energy of a system of particles, U, is the sum of the kinetic energy and potential energy of its constituents. • The internal energy is a state function • It depends only on the values of the state variables • It does NOT depend on the path in the thermodynamic phase space.

5. First Law of Thermodynamics • If some external forces are acting upon the system by virtue of its interaction with the environment then there will be a change in the internal energy. • If the system is an open system, then energy will be exchanged between the system and the environment in the form of heat. • This implies that the change in internal energy should be equal to the work done on the system plus the heat added into the system

6. First Law of Thermodynamics • The first law of thermodynamics states that the internal energy of a system can be changed by doing work on it or by heating/cooling it. • when energy is transferred INTO the system • when work is done BY the system ON the environment.

7. Heat Capacities of Gases • By definition, the heat capacity of a gas is . • Two heat capacities are of interest • CV: Heat capacity at constant volume • CP: Heat capacity at constant pressure • Using the first law of thermodynamics we have

8. Heat Capacities of Gases • Applying this to an ideal gas, we note that by kinetic theory the internal energy of an ideal gas is given by • Thus, for an ideal gas, the internal energy depends ONLY on the temperature • It follows that

9. Adiabatic Processes • A process in which no energy is transferred to a system except by work is said to be adiabatic and the system is then said to be thermally isolated. • For an ideal gas undergoing an adiabatic process, it can be shown that the adiabatic equation of state is given by • The adiabatic equation of state leads to the expression for work

10. Adiabatic Processes • The amount of work needed to change the state of a thermally isolated system depends only on the initial and final states. • The work flowing out of the gas comes at the expense of its thermal energy, implying that its temperature will decrease

11. Potential Temperature • The potential temperatureθ is defined as that temperature the system would assume were it compressed or expanded adiabatically to a reference pressure. • Using the adiabatic equation of state, we have • θ is invariant along an adiabatic path in thermodynamic phase space. • Potential temperature can be used as a tracer during adiabatic processes and can be used to identify diabatic processes (i.e. processes that involves exchanges with heat). • Using the first law of thermodynamics, it can be shown that

12. Thermodynamic Cycles • A thermodynamic cycle is a process in which the thermodynamic system periodically returns to its original macrostate. • Any system undergoing a cyclic process will either do work on the environment or have work done upon it by the environment. • If work is done by the cycle then the energy for the work done must be extracted from some external source.

13. Heat Engines and Refrigerators • We define a heat engine as any thermodynamic cycle which extracts energy from a reservoir in the form of heat and performs mechanical work. • We define a refrigerator as any cyclic thermodynamic process which transfers energy in the form of heat from a reservoir at a lower temperature to a reservoir at a higher temperature.

14. Heat Engines and Refrigerators • The efficiency of a heat engine is defined as • The efficiency of a refrigerator is defined as

15. Carnot Cycle

16. Carnot Cycle: Stage 1 to Stage 2 • The expansion is isothermal because the gas is continuously in thermal equilibrium with the reservoir • The internal energy of the gas remains constant and it absorbs energy , converting all of it into useful work

17. Carnot Cycle: Stage 2 to Stage 3 • Because no energy enters or leaves the system by heat, the work done by the gas is at the cost of its internal energy, causing its temperature to decrease. • Continue the process of expansion until the temperature reaches • The change in internal energy is given by

18. Carnot Cycle: Stage 3 to Stage 4 • Work is done on the gas at constant temperature. • Because the internal energy is held constant, energy in the form of heat, , is expelled from the gas. • By the first law, we have

19. Carnot Cycle: Stage 4 to Stage 1 • No energy is allowed to enter or leave the gas by heat, so the work done on the gas increases its internal energy and therefore its temperature. Thus,

20. Carnot Cycle: Efficiency • To calculate the efficiency of the Carnot engine, we need to compare the total work done by the cycle to the energy absorbed at the high temperature reservoir. • The efficiency is given by

21. The Otto Cycle

22. The Otto Cycle • The work done by the cycle is • Using the ideal gas law, it can be shown that

23. The Otto Cycle • Heat is absorbed during an isochoric process and therefore • The efficiency of the engine is • The efficiency can also be written in terms of the compression ratio

24. The Otto Cycle • Observe that the efficiency of the idealized cycle depends only on the compression ratio • The larger the compression ratio, the greater the efficiency. • Excessive compression ratios can cause the mixture to burn prematurely, causing the engine to actually lose power and efficiency.

25. The Diesel Cycle

26. The Diesel Cycle • The heat absorbed during the cycle is • The heat expelled during the cycle is • The efficiency of the Diesel cycle is

27. The Diesel Cycle • The efficiency of the Diesel cycle can also be given in terms of the compression ratio and the cut-off ratio

28. The Brayton Cycle

29. The Brayton Cycle • Energy is absorbed as heat during combustion (2 to 3) • Energy is released during the isobaric cooling • The efficiency of the engine is

30. The Brayton Cycle • Using the adiabatic equation of state, we have • Note that is the atmospheric temperature and represents the temperature at the compressor exit.