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Adiabatic Work Interactions

Adiabatic Work Interaction• An interaction involving two closed systems for which the events

occurring in both systems can be repeated in such a way that the sole effect external to one system could be duplicated by the rise (or fall) of weights in a standard gravitational field and the sole effect external to the other system can be duplicated by an equivalent fall (or rise) of weights of equal magnitude

Adiabatic Work Interactions

Adiabatic Work Interactions

Consider the following example …

Compare system A+B with system C• This is an adiabatic work interaction – system A+B can be replaced

by a drum that has a weight attached to it

Compare system A with system B+C• External to A – sole effect is lowering a weight (final temperature

of water in system B will be 0 oC)• External to B+C – our experience tells us that it is impossible to

devise an experiment for which the sole effect is the rise of a weight

Adiabatic Work InteractionsTake Home Message

• An adiabatic work interaction requires that all common boundaries be adiabatic walls

• Systems and boundaries must be carefully delineated

Example 3.1Consider the situation illustrated below, in which an electric generator is operated by a falling weight and in which the power generated is dissipated in a resistor. Neglect any dissipative processes such as i2R line losses, friction in bearings, etc. Is this an adiabatic work interaction?

EnergyPostulate III

For any states (1) and (2), in which a closed system is at equilibrium, the change of state represented by (1) (2) and/or the reverse change (2) (1) can occur by at least one adiabatic process and the adiabatic work interaction between this system and its surroundings is determined uniquely by specifying the end states (1) and (2)

• From Postulate III it follows that all stable states can be bridged by adiabatic processes originating from a given initial state

• The adiabatic work for a process is a function of the end states only• This indicates that the adiabatic work is a derived property, which

we give the name total energy, E• The adiabatic work for a given process is given by the total energy

changeaWEE 2112

Energy

Internal Energy• For simple systems (no external force fields or inertial forces) the

total energy is reduced to the internal energy, denoted by U• The internal energy is related to molecular motions, intramolecular

effects, and intermolecular interactions• Postulate I tells us that the internal energy is a function of two

independently variable properties (say and P) plus the masses Mi

• The internal energy is also first order in the total mass of the system

1 2, , , , , nU f P M M M

1 2 1 2, , , , , , , , , ,n naU P M M M U P aM aM aM

Heat Interactions

Heat• The “missing work” for any process (adiabatic or non-adiabatic)• The difference of the total energy change and the actual work

performed

Sign Convention• Work (W) – positive if work is done on the system by the

surroundings• Heat (Q) – positive if heat is “added” to the system• This is the “modern” sign convention• A positive heat or work interaction leads to an increase in the total

energy of the system

WEEQ initialfinal

The First Law for Closed Systems

Energy Balance• Based on our previous findings, we can write

• The left hand side represents the total energy change of the system• The terms on the right hand side (Q and W) represent mechanisms

for energy transfer• Q and W are defined only in terms of the interactions at boundaries

for a prescribed process – “boundary phenomena”

• In first law for closed systems in differential form is

• Differential operators: d for state functions, for path functions

WQE

WQEd

Heat Interactions

Heat Interactions• Consider the following system

• The only type of interaction that can occur between system A and B is a pure heat interaction (W = 0 EA = – EB or QA = – QB)

• If an interaction occurs, the primitive properties of A and B will change

• No interaction: composite system is at equilibrium (Postulate II)• Interaction: the system must tend toward equilibrium and the

interaction must eventually cease (Postulate II)• When the interaction ceases, the systems are said to be in thermal

equilibrium

Heat InteractionsPostulate IV

If the sets of systems A-B and A-C each have no heat interactions when connected across nonadiabatic walls, there will be no heat interaction if systems B and C are also connected(Zeroth Law of Thermodynamics)

Thermometric Temperature• This postulate is used to rank thermometric temperature• We say that if EA = – EB < 0 or equivalently if QA = – QB < 0,

then A > B

• In words – if energy is transferred from system A to B as a result of a pure heat interaction, then the thermometric temperature of system A is greater than that of system B

• For a pure heat interaction to occur, there must be a temperature difference between system A and B

The Ideal Gas

Ideal Gas Review• Ideal gas temperature …

measure the volume of apiston-cylinder devicecontaining helium when placed in both an ice waterand boiling water bath

• Equation of state: PV = NRT• The internal energy and enthalpy of an ideal gas are functions of

temperature only

U = g (T) and H = f (T)

The Ideal Gas

Example 3.3

Two well-insulated cylinders are placed as shown below. The pistons in both cylinders are of identical construction. The clearances between piston and wall are also made identical in both cylinders. The pistons and the connecting rod are metallic.

Cylinder A is filled with gaseous helium at 2 bar and cylinder B is filled with gaseous helium at 1 bar. The temperature is 300 K and the length L is 10 cm. Both pistons are only slightly lubricated.

The stops are removed. After all oscillations have ceased and the system is at rest, the pressures in both cylinders are, for all practical purposes, identical.

Assuming the gases are ideal with a constant Cv and, for simplicity, assuming that the masses of cylinders and pistons are negligible, what are the final temperatures?

The First Law for Open Systems

Open Systems• Consider the system bounded by

the -surface to the right• Over a short time interval a small

quantity of mass nin enters the system• Now consider the composite system+ nin as a closed system and apply the first law

• Rearranging and differentiating gives

• Generalization to multiple streams and integration gives

ininininin nVPWQnEEE 12

inininin nVPEWQEd

out

outoutoutoutin

inininin nVPEnVPEWQE

The First Law for Open Systems

First Law for Open Simple Systems• For simple systems, the total energy E becomes the internal energy U• By defining the enthalpy, H ≡ U + PV, we can write the first law for

open simple systems in differential form as

First Law for Open Non-Simple Systems• If we assume that the total energy can be split into three major parts:

kinetic energy (mv2/2), potential energy (mgz), and internal energy, a generalized first law for open systems can be expressed as

out

outoutin

inin nHnHWQUd

outout

outoutout

inin

ininin n

vgzHn

vgzHWQEd 22

22

2

2vm

zmgUdEdwith

Example 3.5

A 4-m3 storage tank containing 2 m3 of liquid is about to be pressurized with air from a large, high-pressure reservoir through a valve at the top of the tank to permit rapid ejection of the liquid. The air in the reservoir is maintained at 100 bar and 300 K.

The gas space above the liquid contains initially air at 1 bar and 280 K. When the pressure in the tank reaches 5 bar, the liquid transfer valve is opened and the liquid is ejected at the rate of 0.2 m3/min while the tank pressure is maintained at 5 bar.

What is the air temperature when the pressure reaches 5 bar and when the liquid has been drained completely?

Neglect heat interactions at the gas-liquid and gas-tank boundaries. It may be assumed that the gas above the liquid is well mixed and that air is an ideal gas with a constant Cv = 20.9 J/mol-K

Problem 3.6

Problem 3.6