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Chepter 2. The First Law : the concepts
The basic concepts.
system
: the part of the world in which we have a special interest
surroundings : around the system in which we make observation.
according to the types of Boundary
open system
: matter can be transfered through the boundary.
closed system : matter cannot be transfered through the boundary.
Isolated system : a closed system that has neither mechanical nor thermal
contact with surroundings.
Fig 2-1 (a) An open system can exchange matter and
energy with its surroundings. (b) A closed system can
exchange energy with its surroundings, but it cannot
exchange matter. (c) An isolated system can exchange
neither energy nor matter with its surroundings.
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2.1 Work, heat and energy
Work is done when an object is moved aganist an opposing force.
It is eqivalent to change in the height of a weight somewhere in
the surroundings.
Example of doing work : (a) the expansion of a gas.
(b) a chemical reaction
The energy of a system is its capacity to do work
The energy of the system is increased.
( work is done on an otherwise isolated system)
The energy of the system is reduced.
(the system does work)
Fig 2-5 When a system does
work, it stimulates orderly
motion in the surroundings. For
instance, the atoms shown here
may be part of a weight that is
being raised. The ordered
motion of the atoms in a falling
weight does work on the
system.
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Molecular Interpretation 2.1
Thermal motion - the disorderly, random motion of the molecules
In molecular terms
The process of heating is the transfer of energy that makes use of the
difference in thermal motion between the system and the surroundings.
○ When heating the system, the molecules in the system are stimulated to move
more enegetically and energy of the system is increased.
○ When a system heats its surroundings, molecules of the system stimulate the
thermal motion of the molecules in the surroundings.
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Fig 2-4 When energy is transferred to
the surroundings as heat, the transfer
stimulates disordered motion of the
atoms in the surroundings. Transfer
of energy from the surroundings to
the system makes use of disordered
motion (thermal motion) in the
surroundings.
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The energy of a system changes as a result of a temperature
difference between it and its surroundings we say the energy has
been transferred as heat.
Fig 2-3 (a) When an endothermic process
occurs in an adiabatic system, the
temperature falls; (b) if the process is
exothermic, then the temperature rises. (c)
When an endothermic process occurs in a
diathermic container, energy enters as heat
from the surroundings, and the system
remains at the same temperature; (d) if the
process is exothermic, then energy leaves
as heat, and the process is isothermal.
Fig 2-2 (a) A diathermic system is one
that allows energy to escape as heat
through its boundary if there is a
difference in temperature between the
system and its surroundings. (b) An
adiabatic system is one that does not
permit the passage of energy as heat
through its boundary even if there is a
temperature difference between the
system and its surroundings.
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Diathermic boundary (steel and glass) : boundary that do permit
energy transfer as heat.
Adiabatic boundary : boundary that do not permit energy
transferas heat.
Exothermic process : A process that release energy as heat
(combustion)
Endothermic process : A process that absorb energy as heat
( the vaporization of water )
※ An endothermic process in an adiabatic container results in a
lowering of temperature of the system.
※ An exothermic process taking place in a diathermic container
under isothermal conditions results in energy flowing into the
system as heat.
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○ Work is the transfer of energy that makes use of organized
motion
Weight rising or lowering
→ its atoms move in an organized way
: spring motion
: electric motion
When a system does work,
→ surroundings atoms or electronons move in an organized
way
When work is done on a system,
→ molecules in the surroundings re used to transfer energy to it
in an organized way
Fig 2-5 When a system does
work, it stimulates orderly
motion in the surroundings. For
instance, the atoms shown here
may be part of a weight that is
being raised. The ordered
motion of the atoms in a falling
weight does work on the
system.
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The distinction between work and heat is made in the surroundings
: falling weight(Joul's experiment)
→ increase temp.
⇒ stimulate thermal motion in the system
Work : energy transfer making use of the organized motion of atoms in the
surroundings
Heat : energy transfer making use of the thermal motion in the
surroundings.
Example : compressing gas
A particle in a box.
- energy is quantized
⇒ a particle can posses only certain energies. "energy levels"
Bolzmann distribution,
Ne  Ei
Ni 
q
kT
,
q  eEi
kT
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2.2 The First Law
Internal energy U
: the total energy of a system.
● It is impossible to know the absolute value of U.
● Deal only with changes in U i.e. △U
△U = Uf - Ui
The internal energy is an state function
(a function of the properties that determine the current state of the system )
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Molecular Interpretation 2.2
The kinetic energy of one atom, of mass m, at a temperature T.
Ek 
1
1
1
mv x2  mv y2  mv z2
2
2
2
The average energy of each term is
1
kT
2
,
k
is the Boltzmann constant, the
total energy of the monatomic gas(only translation mode) is
U m  U m (0) 
3
NkT
2
, or
3
nRT
2
3
RT
2
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U m (0) is the molar internal energy at T=0.
For a nonlinear polyatomic gas(translational and rotational mode), there is an
additional contribution of 3/2 RT arising from the kinetic energy of rotation.
this case, therefore
U m  U m (0)  3 RT
on (x, y, z) coordinate
non-linear, =(3/2)kT
linear, =kT
The expression for the mean energy of an oscillator of frequency  is worked
out by using the quantum mechanical expression for the energy levels and the
Bolzmann distribution
U m  U m (0)  3RT 
N A h
e h / kT  1
RT when
h << kT
Because the potential energy of interaction between the atoms (or molecules) of
a perfect gas is zero
(
U
)T  0
V
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(a) The conservation of energy
observation : The internal energy of the system may be changed either by doing
work on the system or by heating it.
Whereas we may know how the energy transfer occurred, the system is blind to
the mode employed.
Heat and work are equivalent ways of changing a system`s energy.
△U = q + w
First Law of thermodynamics
For isolated system △U=0
⇒ The change in internal energy of a closed system is equal to the energy
that passes through its boundary as heat or work
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(b) The formal statement of the First Law
The work needed to change an adiabatic system from one specified state to another
specified state is the same however the work is done.
In mountain climbing.
The height we must climb between any two
points is independent of the path we take.
h = Af - Ai = △A
( altitude difference )
Wad = Uf - Ui = △U
Fig 2-6 It is found that the same quantity of work must be done on an
adiabatic system to achieve the same change of state even though
different means of achieving that work may be used. This path
independence implies the existence of a state function, the internal
energy. The change in internal energy is like the change in altitude when
climbing a mountain: its value is independent of path.
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(c) The mechanical definition of heat
For a diathermic system, the thermal contact.
same initial state to the same final state as adiabatic ⇒ △U = Wad
q = Wad -W
⇒ q = △U - W
⇒ △U = q + W
Work and heat
interested in infinitesimal change of state
dU = dq + dW
should concern about dq & dw which occur in surroundings
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2.2 Expansion work
(a) The general expression for work
dw = -Fdz
the work done to move an object a distance dz
against an opposing force F.
dw > 0 system is worked from the surroundings
dz < 0
F = mg
dw < 0 system works to the surroundings
dz > 0
F = mg
the work done when the system expands by dV
against a pressure pex is
Fig 2- 7 When a piston of area A moves out through
a distance dz, it sweeps out a volume dV = A dz. The
external pressure, pex, is equivalent to a weight
pressing on the piston, and the force opposing
expansion is F = pex A.
dw = - pex dV
w
Vf

Vi
pex dV
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(b) Free expansion
By free expansion we mean expansion against zero
opposing force.
w = 0 ( when pex = 0)
(c) Expansion against constant pressure
w =  pex
Vf

Vi
dV = - pex(Vf -Vi)
w =  pex V (ΔV = Vf -Vi )
Fig 2-8 The work done by a gas when it
expands against a constant external pressure,
pex, is equal to the shaded area in this
example of an indicator diagram.
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(d) Reversible expansion
A Reversible change is one that can be reversed by an infinitesimal modification of
a variable
⇒ Equilibrium : an infinitesimal change in the conditions in opposite directions
results in opposite changes in its state.
Example : Thermal equilibrium.
Reversible work
dW = -PexdV = -PdV
∴W =
Vf

Vi
( Pex = p )
pex dV= calculating
if we know P vs V relation, we can calculate.
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(e) Isothermal reversible expansion
PV = nRT
Vf
w   nRT 
Vi
Vf
dV
  nRT ln( )
V
Vi
At high T, more work is done for same volume change.
Matching the external pressure to the internal
pressure at each stage ensures that none of the
systems pushing power is wasted ⇒ max. work is
obtained.
Fig 2-9 The work done by a perfect gas when it expands reversibly and isothermally is
equal to the area under the isotherm p = nRT/V. The work done during the irreversible
expansion against the same final pressure is equal to the rectangular area shown
slightly darker. Note that the reversible work is greater than the irreversible work.
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The maximum work available from a system operating between specified initial and
final states and passing along a specified path is obtained when it is operating reversibly.
. quasi-static path
. Formal - hypothetical path
. Infinite slowness ( satisfactory condition )
example. Fe(s) + 2HCl(aq) → FeCl2(S) + H2(g) ↑
50g
a) a closed vessel off fixed volume
b) an open beaker at 298K
soln ) a) dW = 0 → W = 0 ∴ the pressure is changed very highly
b) W = - Pex △V △V= Vf - Vi Vf = nRT/Pex
W = - Pex × (nRT/Pex) = -nRT = -2.2KJ
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2.4 Heat transaction
In general dU = dq + dWe + dWexp
in addition to expansion work
ex) electrical work
at const volume. dWexp = 0
∴ dU = dqv at const volume, no additional work.
⇒ △U = dqv = qv
(a) Calorimetry
adiabatic bomb calorimeter
△T : the change in temperature of the calorimeter
q= c × △T
heat capacity : calorimeter constant.
① electrical work W = I× V × t = q
∴
C
q
T
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qsample
qs tandard

Tsample
Ts tandard

Csample
Cs tandard
(b) Heat capacity
→ heat capacity at constant volume.
U
Cv  (
)V
T
dU = dqv = CvdT
∴ Cv =
dqv
=
dT
(
U
)V
T
→ △ U = Cv△T
( ∵ Cv : independent of T )
At Phase transition : C = ∞
Fig 2-10 A constant-volume bomb calorimeter. The
`bomb' is the central vessel, which is massive enough
to withstand high pressures. The calorimeter (for
which the heat capacity must be known) is the entire
assembly shown here. To ensure adiabaticity, the
calorimeter is immersed in a water bath with a
temperature continuously readjusted to that of the
calorimeter at each stage of the combustion.
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Fig 2-10 The internal energy of a system
increases as the temperature is raised; this graph
shows its variation as the system is heated at
constant volume. The slope of the graph at any
temperature (as shown by the tangents at A and
B) is the heat capacity at constant volume at that
temperature. Note that, for the system illustrated,
the heat capacity is greater at B than at A.
Fig 2-11 The internal energy of a system varies with
volume and temperature, perhaps as shown here by
the surface. The variation of the internal energy
with temperature at one particular constant volume
is illustrated by the curve drawn parallel to T. The
slope of this curve at any point is the partial
derivative (U/T)v.
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2.5 Enthalpy
At const external pressure.
Some of the energy supplied as heat is converted into the
work required to drive back the surroundings.
∴ dU ＜ dq
(a) The definition of enthalpy
Enthalpy
H = U + pV
state function : independent of the path between them.
→ dH = dqp
Fig2-13 When a system is subjected to
constant pressure and is free to change its
volume, some of the energy supplied as heat
may escape back into the surroundings as
work. In such a case, the change in internal
energy is smaller than the energy supplied
as heat.
( at const. p. no additional work )
→ △H = qp
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Justification 2.1
H + dH = U + dU + ( p + dp )( V + dV )
= U + dU + pV + pdV + Vdp + dp dV
= H + dU + pdV + Vdp
→ dH = dU + pdV + Vdp
= dq + Vdp
( ∴ if the system is in mechanical equilibrium with its surroundings. )
at constant P. dp = 0
⇒ dH = dqp
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(b)The measurement of an enthalpy change
Adiabatic flame calorimeter.
measurement of △ T → △ H = CpdT
Bomb calorimeter
measurement of △ T → △U = CvdT
For liq. and solid
pVm is small
△Um ≈ △Hm
Fig 2-14 A constant-pressure flame
calorimeter consists of this element
immersed in a stirred water bath.
Combustion occurs as a known amount
of reactant is passed through to fuel the
flame, and the rise of temperature is
monitored.
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Ex. 2.2 Relating △H and △U
CaCO3(s) 〓 CaCO3(s)
calcite
aragonite
△Um = 0.21kJ,
P = 1.0 bar,
M = ρVm
ρc = 2.71 gcm-3
ρa =
Vmc = 37cm3
2.93 gcm-3
Vma = 34cm3
△H = H(a) - H(c)
= { U(a) + pV(a) } - { U(c) + pV(c) }
p△V = ( 1×105Pa ) × ( 34-37 ) * 10-6m3 = -0.3J
∴ △H-△U= -0.3J → 0.1% of △U
For a perfect gas
H = U + PV = U + nRT
For a gas reaction
△H = △U + △(ngRT )
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Ex. 2.3 Calculating a change in enthalpy
H20(l) → H20(g)
P = 1atm, T = 373.15 K
I = 0.5A, V = 12V, t = 300sec → q = IVt = 1.8kJ
0.798g water = 0.798 / 18 = 4.43 × 10-2 mole
△ng = 1
△H =
1.8kJ
0.0443 mol
= 41 kJmol-1
△Um = △H - RT = 41 - 3 = 38 kJmol-1
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(c) The variation of enthalpy with temperature
Cp = (
U
) p , heat capacity at const. pressure
T
molar heat capacity, Cp,m = Cp/n
dH = CpdT (at const P)
△H = Cp△T = qp
(at constant pressure)
Cp = a + bT + c/T2
Fig 2-15 The slope of a graph of the enthalpy of a
system subjected to a constant pressure plotted
against temperature is the constant-pressure heat
capacity. The slope of the graph may change with
temperature, in which case the heat capacity varies
with temperature. Thus, the heat capacities at A and
B are different. For gases, the slope of the graph of
enthalpy versus temperature is steeper than that of
the graph of internal energy versus temperature,
and Cp,m is larger than CV,m.
The empirical parameters a,b and c are independent
of temperature
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Ex. 2.4 Evaluating an increase in enthalpy with T.
N2, 25℃ → 100℃

H (T2 )
H (T1 )
dH 

T2
T1
c
(a  bT  2 )dT
T
H(T2)-H(T1) = a(T2-T1)+ b(T22-T12)-c(
1 1

)
T2 T1
a=28.58, b=3.77, c=0.5
H(373K) = H(298K) + 2.20kJmol-1
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(d) The relation between heat capacities
In general
∵ System expand when heated at cons.P.
Cp > Cv
⇒ system do work on the surroundings and some of energy
(heat) escapes back to the surroundings.
∴ same q
Cp△T = Cv△T' ∴ △T <△T'
For a perfect gas
Cp - Cv = nR
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2.6 Adiabatic changes
(a) The work of adiabatic change
U = Cv(Tf-Ti) = CvΔT (at const. volume)
the expansion is adiabatic, ΔU=wad
Wad = CvΔT
For adiabatic, reversible expansion
(perfect gas)
VfTfc = ViTic,
Tf = Ti ( Vi )1/ c
Vf
c=
Cv, m
R
Fig 2-16 To achieve a change of state from one temperature and
volume to another temperature and volume, we may consider the
overall change as composed of two steps. In the first step, the
system expands at constant temperature; there is no change in
internal energy if the system consists of a perfect gas. In the
second step, the temperature of the system is increased at constant
volume. The overall change in internal energy is the sum of the
changes for the two steps.
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Justification 2.2
reversible expansion
dw = - pdV, dU = CVdT
- pdV = CVdT /nR
Cv
Cv
dT
dV
  nR
T
V

Tf
Ti
Cv ln(
ln(
Fig 2-17 The variation of temperature as a perfect gas is expanded
reversibly and adiabatically. The curves are labelled with different
values of c = CV,m/R. Note that the temperature falls most steeply for
gases with low molar heat capacity.
V f dV
dT
  nR 
Vi V
T
dT
dV
)   nR ln( )
T
V
dT c
dV
)  ln( )
T
V
C= CV /nR
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(b) Heat capacity ratio and adiabats
pV   const.
γ :
γ >1
Heat capacity ratio  
C p, m
Cv, m
at constant pressure.
 
Cv, m  R
Cv , m
For a monatomic perfect gas, Cv,m= 3/2 R
∴ γ = 5/3
For nonlinear polyatomic gases, Cv,m=3R
∴ γ = 4/3
Fig 2-18 An adiabat depicts the variation of
pressure with volume when a gas expands
reversibly and adiabatically. (a) An adiabat for a
perfect gas. (b) Note that the pressure declines
more steeply for an adiabat than it does for an
isotherm because the temperature decreases in the
former.
Thermochemistry
The study of the heat produced or required by
chemical reactions.
We can measure q, △U, △H, depending on the
conditions.
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2.7 Standard enthalpy change
Standard state - pure form at 1 bar, at a specified Temp. ( 298.15 K = T )
△vapH° : standard enthalpy of vaporization
H20(l) → H20(g)
△vapH ° = +40.66kJmol-1 at 373K
(a)Enthalpies of physical change
△trsH ° : standard enthalpy of transition
△fusH ° H20(s) → H20(l)
△fusH °(273K)= +6.01kJmol-1
△subH ° C(s,graphite) → C(g)
△subH °(T) =
+716.68kJmol-1
△subH °(T) = △fusH °(T)+ △vapH ° Ho (T)
△solnH ° - limiting enthalpy of solution.
The interactions between the ions are negligible
HCl(g) → HCl(aq) △solnH ° = -75.14 kJmol-1
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(b) Enthalpies of chemical change
The standard reaction enthalpy, ΔrH °
ΔrH ° = -890kJmole-1
CH4(g) + 2 O2(g) = CO2 (g) + 2H2O(l)
The combination of a chemical equation of a chemical equation and a standard
reaction enthalpy is called a thermochmical equation
Consider the reaction
2A + B → 3C + D
The Standard enthalpy ΔrH ° =
 H

m
product
ΔrH ° =
 H

 H

m
reactant

m
(J )
J
(c) Hess's law
The standard enthalpy of an overall reaction is the sum of the standard enthalpies of
the individual reactions into which a reaction may be divided.
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2.8 Standard enthalpies of formation
The standard reaction enthalpies for the
formation of the compound from its elements
in their reference states.
Reference states :
Its most stable state
at the specified Temp. and 1 bar.
(a) The reaction enthalpy in terms of
enthalpies of formation
ΔrH ° =
 H
product

m

 H

m
reactant
36
(b) Group contributions
Mean bond enthalpies : the enthalpy change associated with the breaking of a
specific A-B bond.
Thermochemical groups : an atom of physical group of atoms bond to at least
two other atoms.
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(c) enthalpies of formation and molecular modelling
difficult to estimate standard enthalpies of formation of conformational isomers.
equitorial (8) & axial (9) conformers of methylcyclohexane have different standard
enthalpies of formation even though they consist of the same thermochemical groups
→ the steric repulsions in the axial conformer
⇒ raise its energy relative to that of the equitorial conformer
- the range of their conformational energy difference : 5.9 ~ 7.9 kJmol-1
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2.9 The temperature dependence of reaction enthalpies
Standard reaction enthalpies at different temperatures
may be estimated from heat capacities and the
reaction enthalpy at some other temperature.
T2
H(T2) = H(T1) +
 C dT
  C dT
p
T1
ΔrH°(T2)=ΔrH° (T1) +
T2
T1
r

p
Kirchhoff's law
ΔrCp°=
 C
product

p,m

 C

p ,m
reactant
Fig 2-19 An illustration of the content of Kirchhoff's law. When the
temperature is increased, the enthalpies of the products and the reactants
both increase, but may do so to different extents. In each case, the change
in enthalpy depends on the heat capacities of the substances. The change in
reaction enthalpy reflects the difference in the changes of the enthalpies.
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