Atmospheric Thermodynamics The Second Law of Thermodynamics and Entropy The first law of thermodynamics is a statement of conservation of energy. The second law of thermodynamics is concerned with the maximum fraction of a quantity of heat that can be converted into work. The Carnot Cycle Cyclic process - a series of operations by which the state of a substance (called the working substance) changes but the substance is finally returned to its original state in all respects Work (w) is done by (or on) the working substance if its volume changes. The internal energy (u) of the working substance is unchanged by the cyclic process, since internal energy is a property of state and the initial and final state are the same. Then from the first law of thermodynamics: € q − w = u2− u1 we find: € q − w = 0q = w The net heat absorbed (q) is equal to the work done (w) during the cyclic process (or cycle). Reversible transformation - each state of the system is in equilibrium so that a reversal in the direction of an infinitesimal change returns the working substance and the environment to their original states. Heat engine - a device that does work through the agency of heat.Consider a cycle of a heat engine in which Q1 heat is absorbed and Q2 heat is rejected. The net heat absorbed is: q = Q1 - Q2 and the work (w) done by the engine is: € w = q = Q1− Q2 The efficiency of the engine (η) is defined as: € η=Work done by the engineHeat absorbed by the working substance=wQ1=Q1− Q2Q1 Consider an ideal heat engine as illustrated below: Y - cylinder B - conducting base P - frictionless piston S - nonconducting stand H - infinite warm reservoir C - infinite cold reservoir T1 > T2 Heat is supplied to the working substance when the cylinder (Y) is placed on the warm reservoir (H). Heat is extracted from the working substance when the cylinder (Y) is placed on the cold reservoir (C). Work is done by the working substance when it expands and the piston (P) is pushed outward. Work is done on the working substance when the piston (P) pushes inward and the working substance contracts.Carnot Cycle i. Start at point A with T=T2 The cylinder is placed on the stand and the working substance is compressed (move from A to B). Since the cylinder is on a nonconducting stand no heat is added to or extracted from the working substance, and the compression is adiabatic. During this adiabatic compression the temperature of the working substance increases to T1 Why does the temperature increase during this step of the cycle? ii. The cylinder is placed on the warm reservoir and isothermally expands at temperature T1` (move from B to C). During this step the working substance does work (by expanding against the force of the piston) and extracts a quantity of heat (Q1) from the warm reservoir. How do we know that the working substance extracts heat during this step? iii. The cylinder is placed on the nonconducting stand and expands adiabatically (move from C to D). The temperature decreases from T1 to T2.The working substance does work during this expansion. How do we know that work is done by the working substance for this transformation? iv. The cylinder is placed on the cold reservoir and is compressed isothermally (at temperature T1) back to its original state (move from D to A) During this isothermal compression the working substance rejects a quantity of heat (Q2) to the cold reservoir. How do we know that heat is rejected to the cold reservoir during this step? The amount of work done during this Carnot cycle is given by: € W = pdVV1V2∫ and is given by the area enclosed by ABCD on the p-V diagram. Since this is a cyclic process, the internal energy of the system is unchanged, and the work done is equal to net heat added to the system (Q1 - Q2). The efficiency of this heat engine is given by: € η=Q1− Q2Q1 In this process the heat engine did work by transferring heat from the warm reservoir to the cold reservoir. One statement of the second law of thermodynamics is “only by transferring heat from a warmer to a colder body can heat be converted into work in a cyclic process”Carnot’s theorems For a given range of temperatures no engine can be more efficient than a reversible engine. All reversible engines, working between the same range of temperatures, have the same efficiency. For the Carnot cycle the ratio of heat absorbed (Q1) to heat rejected (Q2) is equal to the ratio of the temperature of the warm reservoir (T1) to the temperature of the cold reservoir (T2): € Q1Q2=T1T2 What are some examples of real heat engines? What is an atmospheric example of a heat engine? If the Carnot cycle is run in reverse a quantity of heat (Q2) is taken from the cold reservoir and a quantity of heat (Q1) is transferred to the warm reservoir (Q1>Q2). In order for heat to be transferred from the cold to the warm reservoir mechanical work (=Q1 - Q2) must be done on the working substance. What is an example of a reversed heat engine? Another statement of the second law of thermodynamics is “heat cannot of itself (i.e. without the performance of work by some external agency) pass from a cooler to a warmer body in a cyclic process”Entropy Consider a portion of a Carnot cycle associated with the isothermal transition from adiabat θ1 to θ2: Is heat added to or extracted from the working substance for this transition? The heat associated with this transition (Qrev) depends on the temperature. If we consider another isothermal transition from adiabat θ1 to θ2 at a different temperature the amount of heat associated with the transition will be different but the ratio € QrevT will be the same. We can then use € QrevT as a measure of the difference between the two adiabats, which is referred to as the difference in entropy. The change in entropy of a system is defined by: € dS ≡dQrevT and the change in entropy for a unit mass of a substance is given by: € ds ≡dqrevT Entropy is a function of state of the system and is independent of the path by which the system is brought to that state. For the transition of a system from state 1 to state 2: € s2− s1=dqrevT12∫Taking the first law of thermodynamics: € dq = du + pdα and the definition of entropy gives: € Tds = du + pdα which is a form of the first law of thermodynamics that contains only functions of state. We can relate entropy and potential