ChemicalChemical ThermodynamicsThermodynamics2013/20142013/2014

3rd Lecture: Work, Heat and the First Law of ThermodynamicsValentim M B Nunes, UD de Engenharia

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IntroductionIntroduction

As we saw before, thermodynamics it’s a science that studies energy transformations but, as we will see, thermodynamics describes macroscopic properties of equilibrium systems.

Although everybody as the feeling of knowing what is energy, it is very difficult to give a precise definition. For our purposes Energy can be defined as the ability to cause changes or realize Work.One of the fundamental laws of nature is the law of conservation of energy. Energy in a system may take on various forms (e.g. kinetic, potential, heat, light) but energy may neither be created nor destroyed.

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Some basic conceptsSome basic concepts

Thermodynamic System – it’s a part of the Universe that is being studied.Exterior or Surroundings of the system – all the rest of the Universe Boundary of the system – its what divides the system from the rest of the universe.

System

Surroundings

Universe = System + Surroundings

Types of systemsTypes of systems

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Isolated System – An isolated system is that in which no transfer of mass & energy takes place across the boundaries of systemClosed System - A closed system in which no transfer of mass takes place across the boundaries of system but energy transfer is possible

Open System - An open system is one in which both mass & energy transfer takes place across the boundaries

Describing Systems Describing Systems

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To describe a given system we need to indicate the components of the system, their physical state (gas, liquid, solid, mixtures) and the state properties, like pressure, p, volume, V, number of moles, n, mass, m, and temperature, T.

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Properties of a systemProperties of a system

When one system suffers a transformation it goes from an initial state to a final state. The properties of the system that are univocally determined by the sate of the system are state functions (or state variables or state properties).

These properties may be either intensive or extensive. Extensive properties depends on the size or extension of the system, like the volume, V. Intensive properties are independent of the size of the system, like temperature or pressure.If we divide an extensive property by the number of moles we obtain an intensive property, like the molar volume:

nVVm

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State of a System at EquilibriumState of a System at Equilibrium

The state of equilibrium is defined by the macroscopic properties and is independent of the history of the system.

Cooling Heating

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Change of stateChange of stateA process or transformation is a change in the state of the system over time, starting with a definite initial state and ending with a definite final state.

There are many types of processes to change the state of a system - at constant volume (isochoric), at constant pressure (isobaric), at constant temperature (isotherm) and so one..

The process is defined by a path, which is the continuous sequence of consecutive states through which the system passes, including theinitial state, the intermediate states, and the final state

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An infinitesimal change of the state function X is written dX. The mathematical operation of summing an infinite number of infinitesimal changes is integration, and the sum is an integral. The sum of the infinitesimal changes of X along a path is a definite integral equal to X:�

If dX obeys this relation—that is, if its integral for given limits has the same value regardless of the path—it is called an exact differential. The differential of a state function is always an exact differential

XXXdXX

X

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Infinitesimal ChangesInfinitesimal Changes

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CyclesCycles

A cyclic process is a process in which the state of the system changes and then returns to the initial state. In this case the integral of dX is written with a cyclic integral.

Since a state function X has the same initial and final values in a cyclic process, X2 is equal to X1 and the cyclic integral of dX is zero:

0dX

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Internal energyInternal energy

The total energy of a system is the Internal Energy, U. The internal energy is a state function. If a system as an initial energy Ui and after a transformation as a n energy Uf then the variation of internal energy, U is:

if UUU

The internal energy is an extensive property, that is, it depends on the size of the system. It can only be changed by two different modes: Work, W, and Heat, Q, trough the boundary of the system.

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Work and HeatWork and Heat

Heat can be viewed as a disordered way of transferring energy (caused by temperature gradient across the boundary) while work is an order way of transferring energy (lifting a weight for instance)

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The 1The 1stst Law of Thermodynamics Law of Thermodynamics

The internal energy of an isolated system is constant. If the system is closed it can only be transferred by heat flow or work done.

dWdQdU

WQU

In differential form

In integrated form:

dQ and dW are not exact differentials what means that they will depend on the path! So heat and work are path functions, they are associated with a process, not a state.

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The 1The 1stst Law of Thermodynamics Law of Thermodynamics

An equivalent formulation of the first law is the following: the work necessary to change an adiabatic system from one state to another is always the same, no matter the type of work done.

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The 1The 1stst Law of Law of Thermodynamics Thermodynamics

But… AU is the same for all the processes!

UUUdU i

f

if

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Signal conventionSignal convention

Process SignalSystem does work on the surroundings -Surroundings do work to the system +Heat absorbed by the system (endothermic process) +Heat absorbed by the surroundings (exothermic process) -

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WorkWork

ApForceAreaForcep ext

dVpdxApdWDistForceWork

extext ance

dVpdW ext

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Expansion Expansion workworkLet us assume the work done by the expansion of a gas against constant external pressure:

ifext

V

Vext

V

V ext VVpdVpdVpW f

i

f

i

VpW ext

In a free expansion, against the vacuum, the external pressure is null (pext = 0), so W=0.

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Isothermal Perfect gas expansion (1 Isothermal Perfect gas expansion (1 step)step)

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Isothermal Perfect gas expansion (two Isothermal Perfect gas expansion (two steps)steps)

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Isothermal Perfect gas expansion (Isothermal Perfect gas expansion (Infinite Infinite stepssteps))If at each step we have p = pext, we have infinite expansions, and maximum work is delivered to the surroundings! This is obtained using a reversible pathreversible path.

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Considering the reversible expansion of a perfect gas, he have:

f

i

V

Vrev pdVWmax,

dVVnRTW f

i

V

Vrev max,

i

frev V

VnRTW lnmax,

R = 8.314 J.K-1.mol-1

Reversible ProcessReversible Process

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SummarySummary

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ProcessIf Pext = 0

If Pext = constant

If the expansion is reversible

0W

VpW ext

i

f

VV

nRTW ln

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Reversible vs IrreversibleReversible vs Irreversible

(T, p1, V1) (T,p2,V2) (T,p1,V1)expansion compression

Process I: expansion against pext = p2 and compression with pext = p1

01221

121122exp

VVppW

VVpVVpWWW

cycle

compcycle

Work done to System! 0dW

Process II: infinite expansions and compressions with pext = p along the path

0lnln2

1

1

2 VVnRT

VVnRTWcycle

Reversible Process!

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HeatHeatIt’s the quantity flowing between the system and the surroundings that can cause a change in temperature of the system and/or the surroundings. Like work, heat its not a state function!What connects Heat with temperature it’s the Heat capacityHeat capacity, C. Units SI are J.K-1.mol-1.

CdTdq

At constant volume:

VV T

qC

At constant pressure:

pp T

qC

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Heat Capacity at constant volumeHeat Capacity at constant volume

At constant volume, dw = 0 so, from the 1st Law, we can easily obtain:

VV T

UC

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Enthalpy, HEnthalpy, HChemical reactions and many other processes, including biological, take place under constant pressure and reversible pV work. Let us define a new function of state, the enthalpy, defined as:

pVUH Differentiating:

VdpdqdHVdppdVpdVdqdH

VdppdVdwdqdHVdppdVdUdH

At constant pressure: dqdH

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Heat Capacity at constant pressureHeat Capacity at constant pressure

Previous result shows that the enthalpy its equal to the heat in a constant pressure process, and we can finally obtain:

pp T

HC

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Relation between Heat Capacity’s for an ideal gas Relation between Heat Capacity’s for an ideal gas

We can now derive a relation between Cp and Cv for an ideal gas:

VpVp T

UTHCC

But, H = U + pV =U + RT (per mole)

ppVp T

URTUCC

Assume this is equal!

RCC Vp For instance, for an ideal monatomic gas CV =3/2 R , so Cp = 5/2 R

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The Joule ExperimentThe Joule Experiment

Let us consider the free expansion of a gas, to getT

T VU

Adiabatic, q = 0

Expansion into the vacuum, w = 0

0U

The experiment proofs that, for an ideal gas, U = U(T)!

P~0

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Total differential of UTotal differential of U

If we regard the internal energy function as U =U(V,T) , then the total differential of U comes:

dTTUdV

VUdU

VT

For an ideal gas so dU = CvdT 0

T

T VU

This means that the internal energy of an ideal gas depends only on temperature. As a consequence: ΔU = 0 for all isothermal expansions or compressions of an ideal gas, and

dTCU V

For any ideal gas change of state.