Chapter 15

The Laws of Thermodynamics

15.2 The Zeroth Law of Thermodynamics

If objects A and B are separately in thermal equilibrium

with a third object C, then objects A and B are in thermal

equilibrium with each another.

When two objects are said to be in thermal equilibrium, it

means the two objects are at the same temperature.

15.3 The First Law of Thermodynamics

IAU=Q-W/

W == work done by the system Q == heat energy entering the system is positive AU == change in internal energy of the system.

An adiabatic process is a quasistatic process with zero heat energy transfer, i.e., Q == o.

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15.4 & 15.5 Work in Thermodynamics Processes

A system undergoes a change of state whenever the thermodynamic coordinates (like pressure, volume, temperature, etc) change in any way whatsoever.

1. Chemical equilibrium: when a system does not undergo a change in internal structure such as a transfer of matter from one place to another.

2. Mechanical equilibrium: when a system does not experience a net (unbalanced) force or torque.

3. Thermal equilibrium: when all parts of a system are at the same temperature, and its temperature is the same as that of the surroundings.

When a system is in chemical, mechanical, and thermal equilibrium, then it is said to be in thermodynamic equilibrium.

During a quasistatic process, the system is at all times infinitely near a state of thermodynamic equilibrium.

The work W done!£. a gas as its volume changes from some initial value Vi to some final value Vi ' equals the area under the curve in a PV diagram. That is,

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f

. >­

, f.,.

~f~",,'L'kil'1 'ft-!"'y +r~,.. ,r"... "'f' t' i \' " > •• 'f ~

I r 1\ i , ' h

, if I

Expansion ~the system expands so the system does positive work. The work done by the system is positive since the final volume is greater than the initial volume.

Compression ~ ~the system contracts so the system does negative work. The work done by the system is negative since the final volume is smaller than the initial volume. A particular state of the system is represented by a point on a PV diagram. The work done by a gas as it is taken from an initial state to a final state depends on the path taken between these states. Let's consider the work done.Qy a gas during different quasistatic processes:

(Aj Constant volume: (an isochoric or isovolumetric process)

------}

Figure 15.6 (a) The substance in the chamber is being heated isochorically because the rigid chamber keeps the vol­ Pi --- - I

ume constant. (b) The pressure-volume plot for an isochoric process is a vertical v straight line, The area under the graph is Volume

zero, indicating that no work is done. (b)

3

!P,

~ 0...

(a)

(B) Constant pressure: (an isobaric process)

Figure 15.5 For an isobaric process. a pressure-versus-volume plot is a horizontal straight line, and the work done [W = P(Vf - V.)] is the colored rectangular area under the graph.

(C) Constant temperature: (an isothermal process)

For the specific case of an ideal gas, the equation of state is PV == nRT,

W =nRT m(~)

Hot water at temperature T (al

Vi VI Volume

(b)

Figure 15.8 (a) The ideal gas in the cylinder is expanding isothermally at temperature T. The force holding the

piston in place is reduced slowly, so the expansion occurs quasi-statically. (b) The work done by the gas is given 4

15.6 Molar specific heats of ideal gases

The heat energy necessary to raise the temperature of n moles of an ideal gas from an initial temperature Ti to a final temperature Tfdepends on the path taken between the initial and final states. Since L\T would be the same for all paths, then L\U is the same for all paths. Thus fron1 the first law of thermodynamics, Q = L\U + Wand this would be different for different paths because W is different for different paths. Resolve this issue by defining the molar specific heat Cv and cp:

Cv = molar specific heat at constant volume cp= molar specific heat at constant pressure

modify equation Q = m c L\T and write instead as:

Q = n CV L\T (for an isochoric process) Q = n cp L\T (for an isobaric process)

Cv and cp are related by:

cp=cv+R

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(a) for monatomic ideal gases (He, Ar, Ne):

3 c =-R

v 2

5 c =-R

p 2

(b) for diatomic ideal gases (H2 , N2 , O2 , CO):

5 c =-R

v 2

7 c =-R

p 2

For Ideal gases, the change in internal energy can be expressed as:

Adiabatic processes for an ideal gas

An adiabatic process is a quasistatic process with no heat energy transfer, i.e., Q == o. This occurs when a gas is compressed or expanded very quickly, or during a very slow expansion of a gas that is thennally insulated from its environment.

How are P, V, and T related during an adiabatic process? Answer is:

PVYPVY - constant 2 2

T Vy-l - constan t TVy-

l = T Vy-

1

1 1 2 2

I !nsulating material

CaJ

(b)

Figure 15.9 (a) The ideal gas in the cylinder is expanding adiabatically. The force holding the piston in place is reduced slowly, so the expansion occurs quasi-statically. (b) A plot of pressure versus volume yields the adiabatic curve shown in red, which intersects the isotherms (blue) at the initial temperature T, and the final temperature Tr• The work done by the gas is given by the colored area.

P,

Volume

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15.8 Heat Engines

A heat engine is a mechanical device by whose agency a system is caused to undergo a cycle where QH is larger than Qc and work We~~ is done by the system. A cycle is a series of processes in which a system is brought back to its initial state.

Figure 15.1 a This schematic representation of a heat engine shows the input heat (magnitude = that originates from the hot reservoir, the work (magnitude = iWj) that the engine does, and the heat (magnitude = IQcI) that the engine rejects to the cold reservoir.

QH = heat energy absorbed by the system from the hot reservoir during one cycle.

Qc = heat energy rejected by the system into the cold reservoir during one cycle.

~~~ = work done by the system during one cycle.

Conservation of energy yields

The thermal efficiency e of a heat engine is defined as:

Combining the last two equations gives,

We~~ = IQHI-IQcl so that the thennal efficiency becomes

A heat reservoir is a body of such a large mass that it may absorb or reject an unlimited quantity of heat without suffering an appreciable change in temperature or any other thennodynamic coordinate.

15.9 Carnot Engine (1824) The Carnot engine is a reversible engine operating between two reservoirs at different temperatures. The hot reservoir is maintained at a temperature TH and the cold reservoir is maintained at a temperature Tc . The maximum work that you can get out of an engine operating between two reservoirs is when a Carnot engine operates between these same two reservoirs. For a Camot

engine, it turns out that the heat energies are related to the temperatures of the reservoirs by:

Temperature =Til

(Carnot engine)

Temperature = Tc

Figure 1 5.11 A Carnot engine is a reversible engine in which all input . heat !QHI originates from a hot reservolI at a single temperature TH, and all rejected heat IQd goes into a cold reservoir at a single temperatme Te· The work done by the engine is \WI.

Thus, the thermal efficiency of a Carnot engine also equals

max Tc eCarnot = 1- ­

TH

15.11 Entropy S In all of nature there is a one-way drive toward thermodynamic equilibrium. There is no way to undo these one-way drives. (a) mechanical one-way drive: frictional dissipation of kinetic energy into internal energy as heat.

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(b) thennal one-way drive: thermal energy flows from to cold. (c) chemical one-way drive: chemicals flow from high concentration to low concentration. This is diffusion.

This is the general irreversibility statement of the second law of thermodynamics.

Isolated systems tend toward disorder and entropy is a measure of that disorder.

Figure 15.20 A block of ice is an example of an ordered system relative to a puddle of water.

The change in entropy ~s of a system as the system undergoes a quasistatic process between two equilibrium states is given by

Entropy is a state variable (like pressure, volume, and temperature). Thus M depends only on the properties of the initial and final equilibrium states of the system.

An isentropic process is one for which ~S=O.

The one-way irreversibility of macroscopic processes in nature can be described by an inequality law. There is a quantity 8 called the entropy which has the following properties:

(a) additive: 8 = 81 + 82 +... (b) each 8 i is a function of the thermodynamic state of the

system. It is a function of temperature and volume. (c) The entropy is not conserved. It can only be created,

but never destroyed. Entropy is created whenever a one-way process occurs. Hence,

~Stotal = 0 (for a reversible process, i.e., not a one-universe

way process)

~Stotal > 0 (when a one-way process occurs). universe

Here, Suniverse is the total entropy of the objects involved.

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