Thermal & Kinetic Lecture 16 Isothermal and Adiabatic processes LECTURE 16 OVERVIEW Isothermal and Adiabatic work Is work a function of state? What distinguishes heat from work? Using thermal processes to do work: Intro. to the Carnot cycle

Last time…. Heat, work, and the 1st law. Adiabatic processes and adiabatic work.

Work done by expanding gases: path dependence T, V, and P are functions of state. Is W also a function of state? V Three different pathways: Isothermal (constant temperature) Isobaric (constant pressure) Isochoric (constant volume) P Isotherm at T 1 2 V1 V2 P2 P1

Heat, work and the 1st law: Adiabatic compression and expansion of an ideal gas However, we’ve shown in Section 2 that CP = CV + R (Eqn. 2.45). Hence, we can write the equation above as: Rewriting this equation taking into account the properties of logs: Equation of an adiabatic Using PV = RT again:

PV curve for an adiabatic and an isothermal process At each point (P, V), the adiabatic for an ideal gas has a slope g times that of an isotherm for an ideal gas. Isotherm V PVg = c, where c is a constant. [1] Let f(P,V) = PVg [2] Proof Using expression [2] above, write down an expression for df (Hint: see Section 2 of the notes re. partial derivatives). ?

? ? PV curve for an adiabatic process If PVg = c (where c is a constant) what can you say about df? Using the expression above, write down an expression for the slope of an adiabatic on a PV diagram. ?

PV curve for an adiabatic process but PV = nRT for an ideal gas Write down an expression for the slope of an isotherm on a PV diagram. ? The adiabatic for an ideal gas has a slope g times that of an isotherm.

Adiabatic work At the beginning of the 19th century it was assumed that heat was a substance called caloric which flowed between bodies. Prompted by measurements carried out by Benjamin Thompson, Joule wanted to determine the precise ‘form’ of heat. http://lectureonline.cl.msu.edu/~mmp/kap11/cd295.htm Water stirred by falling weights turning paddle wheel. Water isolated from surroundings by adiabatic walls of container.

Adiabatic work No matter how the adiabatic work was performed, it always took the same amount of work to take the water between the same two equilibrium states (whose temperatures differed by DT) If a thermally isolated system is brought from one equilibrium state to another, the work necessary to achieve this change is independent of the process used. (1st law, conservation of energy). This seems to contradict what was said earlier – it appears that the work done in this case is path independent? Adiabatic work is a function of state – the adiabatic work done is independent of the path DU = Q + W. If Q=0  DU = W

? Adiabatic work Proof (c is a constant) Work done by a gas in expanding adiabatically from a state (P1, V1) to a state (P2, V2) Proof [1] [2] From expressions [1] and [2] above, determine a formula for the work done in terms of VA and VB. ?

Adiabatic work …but: Therefore, we can rewrite the expression for W as:

The distinction between heat and work “If both heat and work increase the internal energy of a system, what is the distinction between the two at the microscopic level?” Heat changes the populations of the energy levels (so have change in entropy because there’s a change in the number of accessible microstates.) Works changes the energies, with the populations staying the same.

Using thermal processes to do work: heat engines Carnot noted that work is obtained from an engine because there are heat sources at different temperatures. Furthermore, he realised that heat could also flow from a hot to a cold body with no work being done. A temperature difference may be used to produce work OR it can be ‘squandered’ as heat. Engine How do we convert thermal energy transfer into useful work? (e.g. a steam engine) How efficient can we make this cycle? QH QL W

The most efficient process: the Carnot cycle In an ideal engine the temperature difference between the two reservoirs should yield the maximum amount of work possible. Carnot realised that this meant that all transfers of heat should be between bodies of nearly equal temperature. The Carnot engine involves reversible processes (these are the most efficient processes in terms of exploiting a temperature difference to do work). TH QH Engine W QL TL Heat supplied from high temp. reservoir: QH Heat rejected into lower temp. reservoir: QL A Carnot engine operates between only two reservoirs and is reversible. All the heat that is absorbed is absorbed at a constant high temperature (QH) and all the heat that is rejected is rejected at a constant lower temp. (QL).

The most efficient process: the Carnot cycle Carnot engine is an idealisation. We’ll use an ideal gas as our working substance. Carnot cycle may be constructed from a combination of adiabatic and isothermal compressions and expansions. P A B QH Adiabatic C W Isotherm D QL Animation V