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NACE Reference to the Peabody Book Peabody's CONTROL OF PIPELINE CORROSION, 2nd Edition January 27th, 2007 Revision 1.0 Metric Units Ohm's Law

E IR

I

E R

R

E I

E IR I = Current ( amps ) R = Resistance ( ohms ) E = Voltage ( volts )

I

E = Voltage ( volts ) I = Current ( amps ) R = Resistance ( ohms )

10.00 A 100.00 Ω 1000.00 V

E = R = I =

1000.00 V 100.00 Ω 10.00 A

E = I = R =

1000.00 V 10.00 A 100.00 Ω

E R

E = Voltage ( volts ) R = Resistance ( ohms ) I = Current ( amps )

R

I = R = E =

E I

Area of a Circle

D 2 D A R 4 2 2

2

D = Diameter of Circle ( cm ) R = Radius of Circle ( cm ) A = Area of Circle ( cm² )

D R A

= = =

10.00 cm 5.00 cm 78.54 cm²

Area of Pipe Cross-Section

DO2 DI2 DO DI 2 2 A RO RI 2 2 4 4 2

DO DI RO RI A

2

= Outer Diameter of Circle ( cm ) = Inner Diameter of Circle ( cm ) = Outer Radius of Circle ( cm ) = Inner Diameter of Circle ( cm ) = Area of Pipe Cross-Section ( cm² )

DO DI RO RI A

= = = = =

10.00 cm 9.75 cm 5.00 cm 4.88 cm 3.88 cm²

D L

= = = =

10.00 cm 300.00 cm 9424.78 cm² 0.9425 m²

Surface Area of Cylinder ( not including end caps )

S 2RL DL D = Diameter of Cylinder ( cm ) L = Length of Cylinder ( cm )

S = Surface Area of Cylinder ( cm² ) which is Various Prefixes for Powers of Ten

S S

Series Circuit

RT R1 R2 R3 R1 R2 R3 RT

= Resistance of Resistor # 1 ( ohms ) = Resistance of Resistor # 2 ( ohms ) = Resistance of Resistor # 3 ( ohms ) = Total Resistance ( ohms )

R1 R2 R3 RT

= = = =

100.00 Ω 200.00 Ω 300.00 Ω 600.00 Ω

R1 R2 R3 RT

= = = =

100.00 Ω 200.00 Ω 300.00 Ω 54.55 Ω

E = I = P =

10.00 V 20.00 A 200.00 W

I = R = P =

20.00 A 0.50 Ω 200.00 W

Parallel Circuit

RT

R1 R2 R3 RT

1 1 1 1 R1 R2 R3

= Resistance of Resistor # 1 ( ohms ) = Resistance of Resistor # 2 ( ohms ) = Resistance of Resistor # 3 ( ohms ) = Total Resistance ( ohms )

Power Formula

P EI ( IR) I I 2 R E = Voltage ( volts ) I = Current ( amps ) P = Power ( watts ) I = Current ( amps ) R = Resistance ( ohms ) P = Power ( watts )

OR

Color Code For Resistors

Band A 1st Figure Black 0 Brown 1 Red 2 Orange 3 Yellow 4 Green 5 Blue 6 Purple 7 Gray 8 White 9

Band B 2nd Figure Black 0 Brown 1 Red 2 Orange 3 Yellow 4 Green 5 Blue 6 Purple 7 Gray 8 White 9

Values Band C Multiplier Silver Gold

Band D Tolerance 0.01 Red ±2% 0.1 Gold ±5% Silver ± 10 % 1 None ± 20 % 10 100 1,000 10,000 100,000 1,000,000

Black Brown Red Orange Yellow Green Blue

Example

A = Orange = 3 C = Red = 100 B = Blue = 6 D = Silver = ± 10% Resistance Value = 3600 Ω ± 10%, or between 3240 Ω and 3960 Ω. Resistor Calculator

Band A = Band B = Band C = Band D =

3 6 100 ± 10 %

Orange Blue Red Silver

Resistor Value = or between

3,240 Ω

3,600 Ω and

± 10 % 3,960 Ω

Reference Electrode Conversions Relative Values of Typical Reference Electrodes to Copper-Copper Sulfate Reference Electrode Electrode ( Half - Cell ) Copper - Copper Sulfate ( Cu-Cu SO4 ) Silver - Silver Chloride ( Saturated ) Saturated Calomel Zinc ( Pure Zinc )

Convert a Reading of:

Name CSE Ag-AgCl SCE Zn

Potential ( Volts ) 0.000 -0.050 -0.070 -1.100

-0.567 V

to: Electrode Conversion Table

CSE Reference Ag-AgCl Electrode SCE Zn

CSE -0.567 V -0.617 V -0.637 V -1.667 V

Converted To Electrode: Ag-AgCl SCE -0.517 V -0.497 V -0.567 V -0.547 V -0.587 V -0.567 V -1.617 V -1.597 V

Zn 0.533 V 0.483 V 0.463 V -0.567 V

Sacrificial Anode Efficiency and Utilization Factor

Magnesium :

LM

0.256WEU F I

W = Weight of Anode ( kg )

W

E

E

UF I

LM

= Current Efficiency = Utilization Factor = Current ( amps ) = Life of Magnesium Anode ( yrs )

Zinc :

LZ

UF I

LM

W

E

E

UF I

LZ

Anode Material¹

Consumption Rate²

Magnesium (H-1 Alloy) Magnesium (High Potential) Zinc

( kg/amp-yr ) 6.80 - 15.86 7.25 - 8.62 10.75

109.00 kg 0.50 0.85 2.0 A 5.93 yrs

= = = = =

109 kg 0.90 0.85 2.0 A 3.90 yrs

0.0935WEU F I

W = Weight of Anode ( kg ) = Current Efficiency = Utilization Factor = Current ( amps ) = Life of Magnesium Anode ( yrs )

= = = = =

UF I

LZ Common Consumption Current Rates Used² Efficiency² ( kg/amp-yr ) 11.34 0.50 7.71 0.50 10.75 0.90

Utilization Factor³ 0.85 0.85 0.85

¹Anodes installed in suitable chemical backfill. ²Current efficiency with current density. The shown efficiency, and the resulting consumption rate, are at approximately 322.92 mA/m² of anode surface. Efficiencies are higher at higher current densities and lower at lower current densities. ³The Utilization Factor for all Galvanic Anodes is 85% ( 0.85 ).

Dwight's Equation for Single Vertical Anode Resistance to Earth - centimeters

RV

8L 1 ln 2L d

= Soil Resistivity ( ohm-cm )

L = Length of Anode ( cm )

L

d = Diameter of Anode ( cm )

d

Rv = Resistance of Single Vertical Anode to Remote Earth ( ohms )

Rv

10,000 Ω-cm 1200.00 cm 20.30 cm 6.84 Ω

= = = =

Dwight's Equation for Distributed Multiple Vertical Anodes Resistance to Earth - centimeters

Rv

8 L 2L ln 1 ln( 0 . 656 N ) 2NL d S

= Soil Resistivity ( ohm-cm )

L = Length of Anode ( cm )

L

S = Spacing between Anodes ( cm ) N = Number of Anodes d = Diameter of Anode ( cm )

S N d

Rv = Resistance of Multiple Vertical Anodes to Remote Earth ( ohms )

Rv

10,000 Ω-cm 213.36 cm 300.00 cm 10 anodes 20.00 cm 4.57 Ω

= = = = = =

Dwight's Equation for Single Horizontal Vertical Anode (or Pipe) Resistance to Earth - centimeters

4L2 4L S 2 L2 S S 2 L2 RH 1 ln 2L dS L L

= Soil Resistivity ( ohm-cm )

L = Length of Anode ( cm )

L

S = Twice the Depth of Anode ( cm ) d = Diameter of Anode ( cm ) RH = Resistance of Multiple Vertical Anodes to Remote Earth ( ohms )

S d RH

= = = = =

10,000 Ω-cm 213.36 cm 370.00 cm 20.32 cm 22.52 Ω

Dwight's Equation for Distributed Multiple Horizontal Anodes Resistance to Earth - centimeters

RN

R F N

where

F 1

SR

ln( 0.66 N )

= Soil Resistivity ( ohm-cm ) S = Spacing between Anodes ( cm ) N = Number of Anodes

S N

R = Resistance of One Anode ( ohms ) F = Multiple Anode Factor

R F

R N = Resistance of Multiple Vertical Anodes to Remote Earth ( ohms )

RN

= = = = = =

10,000 Ω-cm 0.203 cm 10 anodes 203.20 Ω 146.48 2976.39 Ω

Cable Resistance

RC

RCable LCable 1000

RCable = Resistance per kilometer ( ohm/km) LCable = Length of Cable - Sum of Positive and Negative Leads ( m ) RC = Resistance of Cable ( ohms )

RCable = LCable = RC =

0.833 Ω/km 116.89 m 0.097 Ω

Rectifier Total Circuit Resistance

RT RGbed RC RS RGbed RC RS RT

RGbed RC RS RT

= Ground Bed Resistance ( ohms ) = Resistance of Cable ( ohms ) = Structure/Pipeline-to-Earth Resistance ( ohms ) = Total Circuit Resistance ( ohms )

= = = =

3.08 Ω 0.10 Ω 7.53 Ω 10.71 Ω

Rectifier Efficiency Formula

PAC V AC I AC

PAC

OR

3600 KN T

% Efficiency

PDC VDC I DC K = Meter Constant ( Shown on Face of Meter )

K

N = # of Revolutions of Disk ( Observe for 60 Seconds Minimum )

N =

T = Time of Observation ( sec ) PAC = AC Input Power ( watts )

T = PAC =

0.005 4 10 sec 7.20 W

V DC = DC Volts ( volts ) ADC = DC Amps ( amps ) PDC = DC Output Power ( watts )

VDC = ADC = PDC =

2.00 V 1.00 A 2.00 W

% Efficiency =

27.78 %

% Efficiency = Percent Efficiency

=

PDC 100 PAC

Rectifier Current Output Calculator

I Out Vaccross S factor shunt

Vaccross shunt

S factor I Out

= Voltage Reading Measured Across Shunt ( mV ) = Shunt Factor from Chart ( amp/mV ) = Current Output of Rectifier ( amps )

Vaccross shunt

S factor I Out

= = =

25 mV 0.10 A/mV 2.50 A

Faraday's Law

Wt KIT K I T

Wt

K I T

= Consumption Rate - Electrochemical Equivalent ( kg/amp-yr ) = Current ( amps ) = Time ( yrs ) = Weight Loss of Product ( kg )

= = = =

9.10 kg/A-yr 0.875 A 4.0 yrs 31.9 kg

= = = = = = =

0.34 kg/A-yr 20 yrs 15.00 A 0.60 27.2 kg 6.25 anodes 7 anodes

I Re quired = MD * =

14.70 A 2.50 A

# Anodes = # Anodes =

5.88 anodes 6 anodes

Wt

Consumption Rate K for Various Metals¹ Metal kg/A-yr lb/A-yr Carbon 1.30 2.86 Aluminum 3.00 6.50 Magnesium 4.00 10.00 Iron/Steel 9.10 20.10 High Silicon/Chromium Iron 0.50 1.00 Nickel 9.60 21.20 Copper (Monovalent) 20.80 45.80 Zinc 10.70 28.60 Tin 19.40 42.80 Lead 33.90 74.70 ¹Based on Table 2, Chapter 2, Basic Course manual, Appalachian Underground Corrosion Short Course

Impressed Current -- # of Anodes Required

# Anodes K LDesired I Re quired U Factor WAnode # Anodes

KLDesired I Re quired U FactorW Anode

= Consumption Rate ( kg/amp-yr ) = Desired Lifetime ( yrs ) = Current Required ( amps ) = Utilization Factor = Weight per Anode ( kg ) = # of Anodes to Use rounding up - use

K LDesired I Re quired U Factor WAnode # Anodes # Anodes

# of Anodes Required Based on Current Discharge

# Anodes

I Re quired MD *

I Re quired = Current Required ( amps ) MD * = Maximum Discharge per Anode ( amps ) * From Anode Manufacturer Data

# Anodes = # of Anodes to Use rounding up - use

Coating Resistance Calculations

RCE AS RP / S where

AS DL

RP / S

D L

AS EON TS 1 EOFFTS 1 ETS 1 ION TS 1 IOFFTS 1 I TS 1 EON TS 2 EOFFTS 2 ETS 2 ION TS 2 IOFFTS 2 I TS 2 Eave IC RP / S RCE

E ave IC

ETS 1 ETS 2 ( EOFFTS 1 EON TS 1 ) ( EOFFTS 2 EON TS 2 ) 2 2 I TS 1 I TS 2 ( ION TS 1 IOFFTS 1 ) ( ION TS 2 IOFFTS 2 )

= Diameter of Pipe ( cm ) = Length of Test Span ( m ) = Surface Area of Pipe Test Span ( m² ) = On Voltage Reading - Test Site 1 ( volts ) = Off Voltage Reading - Test Site 1 ( volts ) = IR Drop Voltage Difference - Test Site 1 ( volts ) = On Current Reading - Test Site 1 ( amps ) = Off Current Reading - Test Site 1 ( amps ) = IR Drop Current Difference - Test Site 1 ( amps ) = On Voltage Reading - Test Site 2 ( volts ) = Off Voltage Reading - Test Site 2 ( volts ) = IR Drop Voltage Difference - Test Site 2 ( volts ) = On Current Reading - Test Site 2 ( amps ) = Off Current Reading - Test Site 2 ( amps ) = IR Drop Current Difference - Test Site 2 ( amps ) = Voltage Average over Pipe Test Span ( volts ) = Current over Pipe Test Span ( amps ) = Resistance Pipe - to - Soil ( ohms ) = Effective Coating Resistance of Pipe Test Span ( ohm-m² )

D L

AS EON TS 1 EOFFTS 1 ETS 1 ION TS 1 IOFFTS 1 I TS 1 EON TS 2 EOFFTS 2 ETS 2 ION TS 2 IOFFTS 2 I TS 2 Eave IC RP / S RCE

= = = = = = = = = = = = = = = = = = =

60.96 cm 1609.34 m 3,082.08 m² -2.000 V -0.900 V 1.100 V 3.000 A 0.200 A 2.800 A -1.700 V -0.850 V 0.850 V 2.800 A 0.100 A 2.700 A 0.975 V 0.100 A 9.750 Ω 30,050.27 Ω-m²

2-Wire Current Test over Pipe Test Span

Cathodic Protection Level 2 Training Manual - NACE pg. 4:58

I PS

E PS RPS

where

RPS LPS RP

D = Diameter of Pipe ( cm )

D

t

t

L PS RP R PS E PS I PS

Pipe Size ( in ) 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

= Thickness of Pipe Wall ( cm ) = Length of Pipe Test Span ( m ) = Resistance of Pipe per meter from Table ( µohm/m ) = Resistance of Pipe Test Span ( ohms ) = Measured Voltage Drop over Pipe Test Span ( volts ) = Calculated Current over Pipe Test Span ( amps )

L PS RP R PS E PS I PS

= = = = = = =

76.20 cm 0.953 cm 60.96 m 6.66 µΩ/m 0.00040599 Ω 0.00017 V 0.419 A

Table of Steel Pipe Resistances per Length¹²³ Outside Diameter Wall Thickness Weight Resistance ( in ) ( cm ) ( in ) ( cm ) ( lb/ft ) ( kg/m ) ( µΩ/ft ) ( µΩ/m ) 2.35 0.925 0.154 0.061 3.65 5.43 76.20 256.84 4.50 1.772 0.237 0.093 10.80 16.07 26.80 87.93 6.62 2.606 0.280 0.110 16.00 28.28 15.20 46.87 8.62 3.394 0.322 0.127 28.60 42.56 10.10 33.14 10.75 4.232 0.365 0.144 40.50 60.27 7.13 23.39 12.75 5.020 0.375 0.148 46.60 73.81 5.82 16.09 14.00 5.512 0.375 0.148 54.60 81.26 5.29 17.36 16.00 6.299 0.375 0.148 62.60 93.16 4.61 15.12 18.00 7.087 0.375 0.148 70.60 105.07 4.09 13.42 20.00 7.874 0.375 0.148 78.60 116.97 3.68 12.07 22.00 8.661 0.375 0.148 86.60 128.88 3.34 10.96 24.00 9.449 0.375 0.148 94.60 140.78 3.06 10.04 26.00 10.236 0.375 0.148 102.60 152.69 2.82 6.25 28.00 11.024 0.375 0.148 110.60 164.59 2.62 8.60 30.00 11.811 0.375 0.148 118.70 176.65 2.44 8.01 32.00 12.598 0.375 0.148 126.60 188.41 2.28 7.48 34.00 13.386 0.375 0.148 134.60 200.31 2.15 7.05 36.00 14.173 0.375 0.148 142.60 212.22 2.03 6.66 ¹ Conversions based on 1in = 2.54cm and 1ft = 0.3048m. ² Based on steel density of 489 lbs/ft³ ( 7832 kg/m³ ) and steel resistivity of 18 µΩ/cm. ³ R = 16.061 x resistivity in µΩ/cm = Resistance of 1 ft of Weight per foot of pipe in µΩ.

4-Wire Current Test over Pipe Test Span

Cathodic Protection Level 2 Training Manual - NACE pg. 4:60

I PS K E PS EON E OFF IA K E PS I PS

where

K

= Voltage Drop with Current Applied ( volts ) = Voltage Drop with Current Not Applied ( volts ) = Applied Current to Pipe Test Span ( amps ) = Calibration Factor ( amp/mV ) = Measured Voltage Drop over Pipe Test Span ( volts ) = Calculated Current over Pipe Test Span ( amps )

EON

IA EOFF EON E OFF IA K E PS I PS

= = = = =

0.00508 V 0.00017 V 10.00 A 2.037 A/mV 0.00017 V 0.346 A

Conductance

1 R

C

R

units are in Siemens ( S ) or microSiemens ( µS ) units of Siemen ( S ) =

1 R

= Resistance ( ohms )

C = Conductance ( S )

C

9.750 Ω 0.103 S

= =

Resistivity Calculations * Area may be Cross - Sectional Area of any material ( i.e. Pipe, Soil, Electrolye, etc. ) * For Soil Box Measurements A & L are made to be the same so they cancel out.

RA L Cathodic Protection Level 1 Training Manual - NACE 2000 pg 1:4

A R L

= Area of Cross-Section ( cm² ) = Resistance ( ohms ) = Length ( cm ) = Resistivity ( ohm-cm )

A R L

= = = =

1000.00 cm² 200.000 Ω 3048.00 cm 65.617 Ω-cm

a

= = =

4.00 cm 450.00 Ω 11309.734 Ω-cm

Wenner Method

Cathodic Protection Level 1 Training Manual - NACE 2000 pg 4:28

2aR a

= Spacing of the Pins ( cm )

R = Resistance Measured - shown on Instrument ( ohms )

= Resistivity of Measured Medium ( ohm-cm )

R

Layer Resistivity Calculations

Cathodic Protection Level 2 Training Manual - NACE 2000 pg 4:67

Parallel Resistance Formula

Rc Layer 1

Ra Rb Ra Rb

Layer 2

Layer 3

1avg 2S1 R1

2 avg 2S 2 R2

3avg 2S 3 R3

S1layer S1

S 2layer S 2 S1

S 3layer S 3 S 2

R1layer R1

1layer 2S1layer R1layer

R2layer

R1 R2 R1 R2

2layer 2S 2layer R2layer

S1 = Layer 1 Spacing - Depth Measured ( cm ) R1 = Layer 1 Resistance Measured - shown on Instrument ( ohms )

1avg = Layer 1 Average Resistivity ( ohm-cm )

S1layer = Layer 1 Spacing - Depth Measured ( cm ) R1layer = Layer 1 Resistance Measured - shown on Instrument ( ohms )

1layer = Layer 1 Resistivity ( ohm-cm )

S 2 = Layer 2 Spacing - Total Depth Measured ( cm ) R2 = Layer 2 Resistance Measured - shown on Instrument ( ohms )

R3layer

3layer 2S 3layer R3layer S1 = R1 =

1avg

S1layer R1layer

1layer

2 avg = Layer 2 Average Resistivity ( ohm-cm )

2 avg

S 2layer R2layer

S 3 = Layer 3 Spacing - Total Depth Measured ( cm ) R3 = Layer 3 Resistance Measured - shown on Instrument ( ohms )

3 avg = Layer 3 Average Resistivity ( ohm-cm )

S 3layer = Layer 3 Spacing - Layer Depth Measured ( cm ) R3layer = Layer 3 Parallel Resistance ( ohms )

3layer = Layer 3 Resistivity ( ohm-cm )

= = = =

S2 = R2 =

S 2layer = Layer 2 Spacing - Layer Depth Measured ( cm ) R2layer = Layer 2 Parallel Resistance ( ohms )

2layer = Layer 2 Resistivity ( ohm-cm )

R2 R3 R2 R3

2layer

S3 R3

3 avg

S 3layer R3layer

3layer

159.11 cm 10.00 Ω 9996.90 Ω-cm 159.11 cm 10.00 Ω 9996.90 Ω-cm

= = = =

318.21 cm 7.40 Ω 14795.41 Ω-cm 159.11 cm 28.46 Ω 28452.714 Ω-cm

= = = = = =

477.32 cm 3.10 Ω 9297.12 Ω-cm 159.11 cm 5.33 Ω 5333.230 Ω-cm

3layer

3layer

Shunt Types and Values

Amps

mV

Shunt Value Ohms

5 25 50 1 2 3 4 5 10 15 20 25 30 50 60 75 100

50 25 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

0.01 0.001 0.001 0.05 0.025 0.017 0.0125 0.01 0.005 0.0033 0.0025 0.002 0.0017 0.001 0.0008 0.00067 0.0005

0.1 1 1 0.02 0.04 0.06 0.08 0.1 0.2 0.3 0.4 0.5 0.6 1 1.2 1.5 2

5

50

0.01

0.1

2 0.5 8 25

200 50 80 25

0.1 0.1 0.01 0.001

0.01 0.01 0.1 1

Shunt Rating Holloway Type RS SS SO SW or CP SW or CP SW or CP SW or CP SW or CP SW or CP SW SW SW SW SW SW SW SW J.B Type Agra - Mesa Cott or MCM Red ( MCM ) Red ( Cott ) Yellow ( MCM ) Orange ( MCM )

Shunt Factor A/mV

Rudenberg Formula

y 0.038I Vx log 10 y

y 2 x 2 x

if x > 10y then

0.0052 I x

Vx

= Resistivity of Earth ( ohm-cm )

=

y

y = y =

x I

Vx

= Length of Anode in Earth ( cm ) which is = Distance from Anode ( m ) = Current of Ground Anode ( amps ) = Voltage Potential at Distance x caused by Ground Anode Current ( volts )

x I

Vx

= = =

1000.00 Ω-cm 152.40 cm 1.524 m 21.34 m 2.0 A 0.150 V

Nernst Equation

ne RT a M EE ln nF a M 0

E 0 = Standard State Half - Cell Electrode Potential - from table ( volts )

E0 =

R T

R T

n F

aM

ne

aM E

= Universal Gas Constant = 8.31434 ( J/mol·K ) = Absolute Temperature ( K ) ¹ = Number of Valence Electrons Transferred - from table = Faraday's Constant = 96,485.309 ( C/g·mol ) = Activity of Metal Ions in Solution ² = Activity of the Metal ( a M = 1 for pure metal ) = Electrode Potential in Existing Solution ( volts )

= = n = F = ne aM = aM = E =

-0.763 V 8.31434 J/mol·K 298.15 K 2 96485.31 C/g·mol 0.71 1 -0.827 V

¹ Absolute Temperature is -273.15 ºC. Standard Conditions for pure metals in the emf series are based on one unit activity of metal ions in the electrolyte at 25 ºC with no impurities in the metal or electrolyte and with reference to a standard hydrogen electrode. ² Activity Coefficients can be found in Chemistry or Engineering handbooks at a molar concentration of 0.01 mol. Practical Galvanic Series in Seawater

Half-Cell

Metal

Au/Au++ Pt/Pt++ Cu/Cu++ H2/2H++ Pb/Pb++ Ni/Ni++ Fe/Fe++ Zn/Zn++ Al/Al++ Mg/Mg++

-1.60 to -1.75 -1.10 -1.05 -0.50 to -0.80 -0.20 to -0.50 -0.50 -0.50 -0.20 -0.20 -0.20

Standard Electrode Potential Eº ( volts ) vs. SHE* -1.55 to -1.70 -1.05 -1.00 -0.45 to -0.75 -0.015 to -0.45 -0.45 -0.45 -0.15 -0.15 -0.15

* Standard Hydrogen Electrode ( SHE )

Valence of Common Metals

Metal Aluminum Copper Copper Iron Iron Lead Lead Zinc

Valence 3 2 1 3 2 4 2 2

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

View more...
E IR

I

E R

R

E I

E IR I = Current ( amps ) R = Resistance ( ohms ) E = Voltage ( volts )

I

E = Voltage ( volts ) I = Current ( amps ) R = Resistance ( ohms )

10.00 A 100.00 Ω 1000.00 V

E = R = I =

1000.00 V 100.00 Ω 10.00 A

E = I = R =

1000.00 V 10.00 A 100.00 Ω

E R

E = Voltage ( volts ) R = Resistance ( ohms ) I = Current ( amps )

R

I = R = E =

E I

Area of a Circle

D 2 D A R 4 2 2

2

D = Diameter of Circle ( cm ) R = Radius of Circle ( cm ) A = Area of Circle ( cm² )

D R A

= = =

10.00 cm 5.00 cm 78.54 cm²

Area of Pipe Cross-Section

DO2 DI2 DO DI 2 2 A RO RI 2 2 4 4 2

DO DI RO RI A

2

= Outer Diameter of Circle ( cm ) = Inner Diameter of Circle ( cm ) = Outer Radius of Circle ( cm ) = Inner Diameter of Circle ( cm ) = Area of Pipe Cross-Section ( cm² )

DO DI RO RI A

= = = = =

10.00 cm 9.75 cm 5.00 cm 4.88 cm 3.88 cm²

D L

= = = =

10.00 cm 300.00 cm 9424.78 cm² 0.9425 m²

Surface Area of Cylinder ( not including end caps )

S 2RL DL D = Diameter of Cylinder ( cm ) L = Length of Cylinder ( cm )

S = Surface Area of Cylinder ( cm² ) which is Various Prefixes for Powers of Ten

S S

Series Circuit

RT R1 R2 R3 R1 R2 R3 RT

= Resistance of Resistor # 1 ( ohms ) = Resistance of Resistor # 2 ( ohms ) = Resistance of Resistor # 3 ( ohms ) = Total Resistance ( ohms )

R1 R2 R3 RT

= = = =

100.00 Ω 200.00 Ω 300.00 Ω 600.00 Ω

R1 R2 R3 RT

= = = =

100.00 Ω 200.00 Ω 300.00 Ω 54.55 Ω

E = I = P =

10.00 V 20.00 A 200.00 W

I = R = P =

20.00 A 0.50 Ω 200.00 W

Parallel Circuit

RT

R1 R2 R3 RT

1 1 1 1 R1 R2 R3

= Resistance of Resistor # 1 ( ohms ) = Resistance of Resistor # 2 ( ohms ) = Resistance of Resistor # 3 ( ohms ) = Total Resistance ( ohms )

Power Formula

P EI ( IR) I I 2 R E = Voltage ( volts ) I = Current ( amps ) P = Power ( watts ) I = Current ( amps ) R = Resistance ( ohms ) P = Power ( watts )

OR

Color Code For Resistors

Band A 1st Figure Black 0 Brown 1 Red 2 Orange 3 Yellow 4 Green 5 Blue 6 Purple 7 Gray 8 White 9

Band B 2nd Figure Black 0 Brown 1 Red 2 Orange 3 Yellow 4 Green 5 Blue 6 Purple 7 Gray 8 White 9

Values Band C Multiplier Silver Gold

Band D Tolerance 0.01 Red ±2% 0.1 Gold ±5% Silver ± 10 % 1 None ± 20 % 10 100 1,000 10,000 100,000 1,000,000

Black Brown Red Orange Yellow Green Blue

Example

A = Orange = 3 C = Red = 100 B = Blue = 6 D = Silver = ± 10% Resistance Value = 3600 Ω ± 10%, or between 3240 Ω and 3960 Ω. Resistor Calculator

Band A = Band B = Band C = Band D =

3 6 100 ± 10 %

Orange Blue Red Silver

Resistor Value = or between

3,240 Ω

3,600 Ω and

± 10 % 3,960 Ω

Reference Electrode Conversions Relative Values of Typical Reference Electrodes to Copper-Copper Sulfate Reference Electrode Electrode ( Half - Cell ) Copper - Copper Sulfate ( Cu-Cu SO4 ) Silver - Silver Chloride ( Saturated ) Saturated Calomel Zinc ( Pure Zinc )

Convert a Reading of:

Name CSE Ag-AgCl SCE Zn

Potential ( Volts ) 0.000 -0.050 -0.070 -1.100

-0.567 V

to: Electrode Conversion Table

CSE Reference Ag-AgCl Electrode SCE Zn

CSE -0.567 V -0.617 V -0.637 V -1.667 V

Converted To Electrode: Ag-AgCl SCE -0.517 V -0.497 V -0.567 V -0.547 V -0.587 V -0.567 V -1.617 V -1.597 V

Zn 0.533 V 0.483 V 0.463 V -0.567 V

Sacrificial Anode Efficiency and Utilization Factor

Magnesium :

LM

0.256WEU F I

W = Weight of Anode ( kg )

W

E

E

UF I

LM

= Current Efficiency = Utilization Factor = Current ( amps ) = Life of Magnesium Anode ( yrs )

Zinc :

LZ

UF I

LM

W

E

E

UF I

LZ

Anode Material¹

Consumption Rate²

Magnesium (H-1 Alloy) Magnesium (High Potential) Zinc

( kg/amp-yr ) 6.80 - 15.86 7.25 - 8.62 10.75

109.00 kg 0.50 0.85 2.0 A 5.93 yrs

= = = = =

109 kg 0.90 0.85 2.0 A 3.90 yrs

0.0935WEU F I

W = Weight of Anode ( kg ) = Current Efficiency = Utilization Factor = Current ( amps ) = Life of Magnesium Anode ( yrs )

= = = = =

UF I

LZ Common Consumption Current Rates Used² Efficiency² ( kg/amp-yr ) 11.34 0.50 7.71 0.50 10.75 0.90

Utilization Factor³ 0.85 0.85 0.85

¹Anodes installed in suitable chemical backfill. ²Current efficiency with current density. The shown efficiency, and the resulting consumption rate, are at approximately 322.92 mA/m² of anode surface. Efficiencies are higher at higher current densities and lower at lower current densities. ³The Utilization Factor for all Galvanic Anodes is 85% ( 0.85 ).

Dwight's Equation for Single Vertical Anode Resistance to Earth - centimeters

RV

8L 1 ln 2L d

= Soil Resistivity ( ohm-cm )

L = Length of Anode ( cm )

L

d = Diameter of Anode ( cm )

d

Rv = Resistance of Single Vertical Anode to Remote Earth ( ohms )

Rv

10,000 Ω-cm 1200.00 cm 20.30 cm 6.84 Ω

= = = =

Dwight's Equation for Distributed Multiple Vertical Anodes Resistance to Earth - centimeters

Rv

8 L 2L ln 1 ln( 0 . 656 N ) 2NL d S

= Soil Resistivity ( ohm-cm )

L = Length of Anode ( cm )

L

S = Spacing between Anodes ( cm ) N = Number of Anodes d = Diameter of Anode ( cm )

S N d

Rv = Resistance of Multiple Vertical Anodes to Remote Earth ( ohms )

Rv

10,000 Ω-cm 213.36 cm 300.00 cm 10 anodes 20.00 cm 4.57 Ω

= = = = = =

Dwight's Equation for Single Horizontal Vertical Anode (or Pipe) Resistance to Earth - centimeters

4L2 4L S 2 L2 S S 2 L2 RH 1 ln 2L dS L L

= Soil Resistivity ( ohm-cm )

L = Length of Anode ( cm )

L

S = Twice the Depth of Anode ( cm ) d = Diameter of Anode ( cm ) RH = Resistance of Multiple Vertical Anodes to Remote Earth ( ohms )

S d RH

= = = = =

10,000 Ω-cm 213.36 cm 370.00 cm 20.32 cm 22.52 Ω

Dwight's Equation for Distributed Multiple Horizontal Anodes Resistance to Earth - centimeters

RN

R F N

where

F 1

SR

ln( 0.66 N )

= Soil Resistivity ( ohm-cm ) S = Spacing between Anodes ( cm ) N = Number of Anodes

S N

R = Resistance of One Anode ( ohms ) F = Multiple Anode Factor

R F

R N = Resistance of Multiple Vertical Anodes to Remote Earth ( ohms )

RN

= = = = = =

10,000 Ω-cm 0.203 cm 10 anodes 203.20 Ω 146.48 2976.39 Ω

Cable Resistance

RC

RCable LCable 1000

RCable = Resistance per kilometer ( ohm/km) LCable = Length of Cable - Sum of Positive and Negative Leads ( m ) RC = Resistance of Cable ( ohms )

RCable = LCable = RC =

0.833 Ω/km 116.89 m 0.097 Ω

Rectifier Total Circuit Resistance

RT RGbed RC RS RGbed RC RS RT

RGbed RC RS RT

= Ground Bed Resistance ( ohms ) = Resistance of Cable ( ohms ) = Structure/Pipeline-to-Earth Resistance ( ohms ) = Total Circuit Resistance ( ohms )

= = = =

3.08 Ω 0.10 Ω 7.53 Ω 10.71 Ω

Rectifier Efficiency Formula

PAC V AC I AC

PAC

OR

3600 KN T

% Efficiency

PDC VDC I DC K = Meter Constant ( Shown on Face of Meter )

K

N = # of Revolutions of Disk ( Observe for 60 Seconds Minimum )

N =

T = Time of Observation ( sec ) PAC = AC Input Power ( watts )

T = PAC =

0.005 4 10 sec 7.20 W

V DC = DC Volts ( volts ) ADC = DC Amps ( amps ) PDC = DC Output Power ( watts )

VDC = ADC = PDC =

2.00 V 1.00 A 2.00 W

% Efficiency =

27.78 %

% Efficiency = Percent Efficiency

=

PDC 100 PAC

Rectifier Current Output Calculator

I Out Vaccross S factor shunt

Vaccross shunt

S factor I Out

= Voltage Reading Measured Across Shunt ( mV ) = Shunt Factor from Chart ( amp/mV ) = Current Output of Rectifier ( amps )

Vaccross shunt

S factor I Out

= = =

25 mV 0.10 A/mV 2.50 A

Faraday's Law

Wt KIT K I T

Wt

K I T

= Consumption Rate - Electrochemical Equivalent ( kg/amp-yr ) = Current ( amps ) = Time ( yrs ) = Weight Loss of Product ( kg )

= = = =

9.10 kg/A-yr 0.875 A 4.0 yrs 31.9 kg

= = = = = = =

0.34 kg/A-yr 20 yrs 15.00 A 0.60 27.2 kg 6.25 anodes 7 anodes

I Re quired = MD * =

14.70 A 2.50 A

# Anodes = # Anodes =

5.88 anodes 6 anodes

Wt

Consumption Rate K for Various Metals¹ Metal kg/A-yr lb/A-yr Carbon 1.30 2.86 Aluminum 3.00 6.50 Magnesium 4.00 10.00 Iron/Steel 9.10 20.10 High Silicon/Chromium Iron 0.50 1.00 Nickel 9.60 21.20 Copper (Monovalent) 20.80 45.80 Zinc 10.70 28.60 Tin 19.40 42.80 Lead 33.90 74.70 ¹Based on Table 2, Chapter 2, Basic Course manual, Appalachian Underground Corrosion Short Course

Impressed Current -- # of Anodes Required

# Anodes K LDesired I Re quired U Factor WAnode # Anodes

KLDesired I Re quired U FactorW Anode

= Consumption Rate ( kg/amp-yr ) = Desired Lifetime ( yrs ) = Current Required ( amps ) = Utilization Factor = Weight per Anode ( kg ) = # of Anodes to Use rounding up - use

K LDesired I Re quired U Factor WAnode # Anodes # Anodes

# of Anodes Required Based on Current Discharge

# Anodes

I Re quired MD *

I Re quired = Current Required ( amps ) MD * = Maximum Discharge per Anode ( amps ) * From Anode Manufacturer Data

# Anodes = # of Anodes to Use rounding up - use

Coating Resistance Calculations

RCE AS RP / S where

AS DL

RP / S

D L

AS EON TS 1 EOFFTS 1 ETS 1 ION TS 1 IOFFTS 1 I TS 1 EON TS 2 EOFFTS 2 ETS 2 ION TS 2 IOFFTS 2 I TS 2 Eave IC RP / S RCE

E ave IC

ETS 1 ETS 2 ( EOFFTS 1 EON TS 1 ) ( EOFFTS 2 EON TS 2 ) 2 2 I TS 1 I TS 2 ( ION TS 1 IOFFTS 1 ) ( ION TS 2 IOFFTS 2 )

= Diameter of Pipe ( cm ) = Length of Test Span ( m ) = Surface Area of Pipe Test Span ( m² ) = On Voltage Reading - Test Site 1 ( volts ) = Off Voltage Reading - Test Site 1 ( volts ) = IR Drop Voltage Difference - Test Site 1 ( volts ) = On Current Reading - Test Site 1 ( amps ) = Off Current Reading - Test Site 1 ( amps ) = IR Drop Current Difference - Test Site 1 ( amps ) = On Voltage Reading - Test Site 2 ( volts ) = Off Voltage Reading - Test Site 2 ( volts ) = IR Drop Voltage Difference - Test Site 2 ( volts ) = On Current Reading - Test Site 2 ( amps ) = Off Current Reading - Test Site 2 ( amps ) = IR Drop Current Difference - Test Site 2 ( amps ) = Voltage Average over Pipe Test Span ( volts ) = Current over Pipe Test Span ( amps ) = Resistance Pipe - to - Soil ( ohms ) = Effective Coating Resistance of Pipe Test Span ( ohm-m² )

D L

AS EON TS 1 EOFFTS 1 ETS 1 ION TS 1 IOFFTS 1 I TS 1 EON TS 2 EOFFTS 2 ETS 2 ION TS 2 IOFFTS 2 I TS 2 Eave IC RP / S RCE

= = = = = = = = = = = = = = = = = = =

60.96 cm 1609.34 m 3,082.08 m² -2.000 V -0.900 V 1.100 V 3.000 A 0.200 A 2.800 A -1.700 V -0.850 V 0.850 V 2.800 A 0.100 A 2.700 A 0.975 V 0.100 A 9.750 Ω 30,050.27 Ω-m²

2-Wire Current Test over Pipe Test Span

Cathodic Protection Level 2 Training Manual - NACE pg. 4:58

I PS

E PS RPS

where

RPS LPS RP

D = Diameter of Pipe ( cm )

D

t

t

L PS RP R PS E PS I PS

Pipe Size ( in ) 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

= Thickness of Pipe Wall ( cm ) = Length of Pipe Test Span ( m ) = Resistance of Pipe per meter from Table ( µohm/m ) = Resistance of Pipe Test Span ( ohms ) = Measured Voltage Drop over Pipe Test Span ( volts ) = Calculated Current over Pipe Test Span ( amps )

L PS RP R PS E PS I PS

= = = = = = =

76.20 cm 0.953 cm 60.96 m 6.66 µΩ/m 0.00040599 Ω 0.00017 V 0.419 A

Table of Steel Pipe Resistances per Length¹²³ Outside Diameter Wall Thickness Weight Resistance ( in ) ( cm ) ( in ) ( cm ) ( lb/ft ) ( kg/m ) ( µΩ/ft ) ( µΩ/m ) 2.35 0.925 0.154 0.061 3.65 5.43 76.20 256.84 4.50 1.772 0.237 0.093 10.80 16.07 26.80 87.93 6.62 2.606 0.280 0.110 16.00 28.28 15.20 46.87 8.62 3.394 0.322 0.127 28.60 42.56 10.10 33.14 10.75 4.232 0.365 0.144 40.50 60.27 7.13 23.39 12.75 5.020 0.375 0.148 46.60 73.81 5.82 16.09 14.00 5.512 0.375 0.148 54.60 81.26 5.29 17.36 16.00 6.299 0.375 0.148 62.60 93.16 4.61 15.12 18.00 7.087 0.375 0.148 70.60 105.07 4.09 13.42 20.00 7.874 0.375 0.148 78.60 116.97 3.68 12.07 22.00 8.661 0.375 0.148 86.60 128.88 3.34 10.96 24.00 9.449 0.375 0.148 94.60 140.78 3.06 10.04 26.00 10.236 0.375 0.148 102.60 152.69 2.82 6.25 28.00 11.024 0.375 0.148 110.60 164.59 2.62 8.60 30.00 11.811 0.375 0.148 118.70 176.65 2.44 8.01 32.00 12.598 0.375 0.148 126.60 188.41 2.28 7.48 34.00 13.386 0.375 0.148 134.60 200.31 2.15 7.05 36.00 14.173 0.375 0.148 142.60 212.22 2.03 6.66 ¹ Conversions based on 1in = 2.54cm and 1ft = 0.3048m. ² Based on steel density of 489 lbs/ft³ ( 7832 kg/m³ ) and steel resistivity of 18 µΩ/cm. ³ R = 16.061 x resistivity in µΩ/cm = Resistance of 1 ft of Weight per foot of pipe in µΩ.

4-Wire Current Test over Pipe Test Span

Cathodic Protection Level 2 Training Manual - NACE pg. 4:60

I PS K E PS EON E OFF IA K E PS I PS

where

K

= Voltage Drop with Current Applied ( volts ) = Voltage Drop with Current Not Applied ( volts ) = Applied Current to Pipe Test Span ( amps ) = Calibration Factor ( amp/mV ) = Measured Voltage Drop over Pipe Test Span ( volts ) = Calculated Current over Pipe Test Span ( amps )

EON

IA EOFF EON E OFF IA K E PS I PS

= = = = =

0.00508 V 0.00017 V 10.00 A 2.037 A/mV 0.00017 V 0.346 A

Conductance

1 R

C

R

units are in Siemens ( S ) or microSiemens ( µS ) units of Siemen ( S ) =

1 R

= Resistance ( ohms )

C = Conductance ( S )

C

9.750 Ω 0.103 S

= =

Resistivity Calculations * Area may be Cross - Sectional Area of any material ( i.e. Pipe, Soil, Electrolye, etc. ) * For Soil Box Measurements A & L are made to be the same so they cancel out.

RA L Cathodic Protection Level 1 Training Manual - NACE 2000 pg 1:4

A R L

= Area of Cross-Section ( cm² ) = Resistance ( ohms ) = Length ( cm ) = Resistivity ( ohm-cm )

A R L

= = = =

1000.00 cm² 200.000 Ω 3048.00 cm 65.617 Ω-cm

a

= = =

4.00 cm 450.00 Ω 11309.734 Ω-cm

Wenner Method

Cathodic Protection Level 1 Training Manual - NACE 2000 pg 4:28

2aR a

= Spacing of the Pins ( cm )

R = Resistance Measured - shown on Instrument ( ohms )

= Resistivity of Measured Medium ( ohm-cm )

R

Layer Resistivity Calculations

Cathodic Protection Level 2 Training Manual - NACE 2000 pg 4:67

Parallel Resistance Formula

Rc Layer 1

Ra Rb Ra Rb

Layer 2

Layer 3

1avg 2S1 R1

2 avg 2S 2 R2

3avg 2S 3 R3

S1layer S1

S 2layer S 2 S1

S 3layer S 3 S 2

R1layer R1

1layer 2S1layer R1layer

R2layer

R1 R2 R1 R2

2layer 2S 2layer R2layer

S1 = Layer 1 Spacing - Depth Measured ( cm ) R1 = Layer 1 Resistance Measured - shown on Instrument ( ohms )

1avg = Layer 1 Average Resistivity ( ohm-cm )

S1layer = Layer 1 Spacing - Depth Measured ( cm ) R1layer = Layer 1 Resistance Measured - shown on Instrument ( ohms )

1layer = Layer 1 Resistivity ( ohm-cm )

S 2 = Layer 2 Spacing - Total Depth Measured ( cm ) R2 = Layer 2 Resistance Measured - shown on Instrument ( ohms )

R3layer

3layer 2S 3layer R3layer S1 = R1 =

1avg

S1layer R1layer

1layer

2 avg = Layer 2 Average Resistivity ( ohm-cm )

2 avg

S 2layer R2layer

S 3 = Layer 3 Spacing - Total Depth Measured ( cm ) R3 = Layer 3 Resistance Measured - shown on Instrument ( ohms )

3 avg = Layer 3 Average Resistivity ( ohm-cm )

S 3layer = Layer 3 Spacing - Layer Depth Measured ( cm ) R3layer = Layer 3 Parallel Resistance ( ohms )

3layer = Layer 3 Resistivity ( ohm-cm )

= = = =

S2 = R2 =

S 2layer = Layer 2 Spacing - Layer Depth Measured ( cm ) R2layer = Layer 2 Parallel Resistance ( ohms )

2layer = Layer 2 Resistivity ( ohm-cm )

R2 R3 R2 R3

2layer

S3 R3

3 avg

S 3layer R3layer

3layer

159.11 cm 10.00 Ω 9996.90 Ω-cm 159.11 cm 10.00 Ω 9996.90 Ω-cm

= = = =

318.21 cm 7.40 Ω 14795.41 Ω-cm 159.11 cm 28.46 Ω 28452.714 Ω-cm

= = = = = =

477.32 cm 3.10 Ω 9297.12 Ω-cm 159.11 cm 5.33 Ω 5333.230 Ω-cm

3layer

3layer

Shunt Types and Values

Amps

mV

Shunt Value Ohms

5 25 50 1 2 3 4 5 10 15 20 25 30 50 60 75 100

50 25 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

0.01 0.001 0.001 0.05 0.025 0.017 0.0125 0.01 0.005 0.0033 0.0025 0.002 0.0017 0.001 0.0008 0.00067 0.0005

0.1 1 1 0.02 0.04 0.06 0.08 0.1 0.2 0.3 0.4 0.5 0.6 1 1.2 1.5 2

5

50

0.01

0.1

2 0.5 8 25

200 50 80 25

0.1 0.1 0.01 0.001

0.01 0.01 0.1 1

Shunt Rating Holloway Type RS SS SO SW or CP SW or CP SW or CP SW or CP SW or CP SW or CP SW SW SW SW SW SW SW SW J.B Type Agra - Mesa Cott or MCM Red ( MCM ) Red ( Cott ) Yellow ( MCM ) Orange ( MCM )

Shunt Factor A/mV

Rudenberg Formula

y 0.038I Vx log 10 y

y 2 x 2 x

if x > 10y then

0.0052 I x

Vx

= Resistivity of Earth ( ohm-cm )

=

y

y = y =

x I

Vx

= Length of Anode in Earth ( cm ) which is = Distance from Anode ( m ) = Current of Ground Anode ( amps ) = Voltage Potential at Distance x caused by Ground Anode Current ( volts )

x I

Vx

= = =

1000.00 Ω-cm 152.40 cm 1.524 m 21.34 m 2.0 A 0.150 V

Nernst Equation

ne RT a M EE ln nF a M 0

E 0 = Standard State Half - Cell Electrode Potential - from table ( volts )

E0 =

R T

R T

n F

aM

ne

aM E

= Universal Gas Constant = 8.31434 ( J/mol·K ) = Absolute Temperature ( K ) ¹ = Number of Valence Electrons Transferred - from table = Faraday's Constant = 96,485.309 ( C/g·mol ) = Activity of Metal Ions in Solution ² = Activity of the Metal ( a M = 1 for pure metal ) = Electrode Potential in Existing Solution ( volts )

= = n = F = ne aM = aM = E =

-0.763 V 8.31434 J/mol·K 298.15 K 2 96485.31 C/g·mol 0.71 1 -0.827 V

¹ Absolute Temperature is -273.15 ºC. Standard Conditions for pure metals in the emf series are based on one unit activity of metal ions in the electrolyte at 25 ºC with no impurities in the metal or electrolyte and with reference to a standard hydrogen electrode. ² Activity Coefficients can be found in Chemistry or Engineering handbooks at a molar concentration of 0.01 mol. Practical Galvanic Series in Seawater

Half-Cell

Metal

Au/Au++ Pt/Pt++ Cu/Cu++ H2/2H++ Pb/Pb++ Ni/Ni++ Fe/Fe++ Zn/Zn++ Al/Al++ Mg/Mg++

-1.60 to -1.75 -1.10 -1.05 -0.50 to -0.80 -0.20 to -0.50 -0.50 -0.50 -0.20 -0.20 -0.20

Standard Electrode Potential Eº ( volts ) vs. SHE* -1.55 to -1.70 -1.05 -1.00 -0.45 to -0.75 -0.015 to -0.45 -0.45 -0.45 -0.15 -0.15 -0.15

* Standard Hydrogen Electrode ( SHE )

Valence of Common Metals

Metal Aluminum Copper Copper Iron Iron Lead Lead Zinc

Valence 3 2 1 3 2 4 2 2

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

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