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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 th
e 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 2-2 (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.
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.
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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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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.
○ Work is the transfer of energy that makes use of organized
motion
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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, , q
NeN
kTE
i
i
kTEieq
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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.
The average energy of each term is , is the Boltzmann constant, the
total energy of the monatomic gas(only translation mode) is , or
222
2
1
2
1
2
1zyxk mvmvmvE
RTUU mm 2
3)0(
kT2
1k
NkT2
3nRT
2
3
11
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. t
his case, therefore
The expression for the mean energy of an oscillator of frequency is worked o
ut by using the quantum mechanical expression for the energy levels and the B
olzmann distribution
)0(mU
RTUU mm 3)0( on (x, y, z) coordinate
non-linear, =(3/2)kT
linear, =kT
13)0(
/
kThA
mm e
hNRTUU
RT when << h kT
Because the potential energy of interaction between the atoms (or molecules) of a perfect gas is zero
0)(
TV
U
12
(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 t
o 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 = W⇒ △ ad
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
15
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
dw = - pex dV 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.
dVpwf
i
V
V ex
16
(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)
w = (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.
f
i
V
Vex dVp
Vpex
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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 ( Pex = p )
∴ W = = calculating
if we know P vs V relation, we can calculate.
dVpf
i
V
V ex
18
(e) Isothermal reversible expansion
PV = nRT
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.
)ln(i
fV
V V
VnRT
V
dVnRTw
f
i
19
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 reversibl
y.
. 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= V△ △ f - Vi Vf = nRT/Pex
W = - Pex × (nRT/Pex) = -nRT = -2.2KJ
20
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 ∴T
qC
21
(b) Heat capacity
→ heat capacity at constant volume.
dU = dqv = CvdT
∴ Cv = =
→ △ U = Cv T △
( C∵ v : 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.
dards
sample
dards
sample
dards
sample
C
C
T
T
q
q
tantantan
VT
U)(
Vv T
UC )(
dT
dqv
22
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
( at const. p. no additional work )
→ △H = qp
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.
24
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
25
(b)The measurement of an enthalpy change
Adiabatic flame calorimeter.
measurement of T → H = C△ △ pd
T
Bomb calorimeter
measurement of T → U = C△ △ vdT
For liq. and solid
pVm is small
△Um ≈ H△ m
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.
26
Ex. 2.2 Relating H and U △ △
CaCO3(s) 〓 CaCO3(s)
calcite aragonite
△Um = 0.21kJ,
P = 1.0 bar, ρc = 2.71 gcm-3 ρa = 2.93 gcm-3
M = ρVm Vmc = 37cm3 Vma = 34cm3
△H = H(a) - H(c)
= { U(a) + pV(a) } - { U(c) + pV(c) }
p V = ( 1×10△ 5Pa ) × ( 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 + (n△ △ △ gRT )
27
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 = = 41 kJmol-1
△Um = H - RT = 41 - 3 = 38 kJmol△ -1
mol
kJ
0443.0
8.1
28
(c) The variation of enthalpy with temperature
Cp = , heat capacity at const. pressure
molar heat capacity, Cp,m = Cp/n
dH = CpdT (at const P)
△H = Cp T = q△ p (at constant pressure)
Cp = a + bT + c/T2
The empirical parameters a,b and c are independent
of temperature
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.
pT
U)(
29
Ex. 2.4 Evaluating an increase in enthalpy with T.
N2, 25 → 100 ℃ ℃
H(T2)-H(T1) = a(T2-T1)+ b(T22-T1
2)-c( )
a=28.58, b=3.77, c=0.5
H(373K) = H(298K) + 2.20kJmol-1
dTT
cbTadH
T
T
TH
TH)(
2
1
2
12
)(
)(
12
11
TT
30
(d) The relation between heat capacities
In general
Cp > Cv System expand when heated at cons.P. ∵
⇒ 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
31
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 = ViTi
c, c =
Tf = Ti
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.
R
C mv,
c
f
i
V
V /1)(
32
Justification 2.2
reversible expansion
dw = - pdV, dU = CVdT
- pdV = CVdT /nR
C= CV /nR
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.
f
i
f
i
V
V
T
Tv V
dVnR
T
dTC
V
dVnR
T
dTCv
)ln()ln(V
dVnR
T
dTCv
)ln()ln(V
dV
T
dT c
33
(b) Heat capacity ratio and adiabats
γ : Heat capacity ratio
γ > 1 at constant pressure.
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.
.constpV
mv
mp
C
C
,
,
mv
mv
C
RC
,
,
34
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) = +716.68kJm
ol-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
35
(b) Enthalpies of chemical change
The standard reaction enthalpy, ΔrH °
CH4(g) + 2 O2(g) = CO2 (g) + 2H2O(l) ΔrH ° = -890kJmole-1
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 ° =
ΔrH ° = (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.
reactantproduct
mm HH
)(J
JHm
36
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 ° = reactantproduct
mm HH
37
(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.
38
(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
39
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.
H(T2) = H(T1) +
ΔrH°(T2)=ΔrH° (T1) +
Kirchhoff's law
ΔrCp°=
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.
dTCT
T p2
1
dTCT
T pr 2
1
reactant
,product
,
mpmp CC
