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Approximate   time   required:     3h
It   is   possible   to   analyze   spectrophotometrically   the   phosphate   concentration   of   an   aqueous   solution
(PO
4
3 ‐
(aq)).
In   an   acidic   environment,   ammonium   molybdate   and   antimonyl   tartrate   combine   with   any   phosphate   present   as   PO
4
3 ‐
(aq)   to   form   an   antimony ‐ phosphate ‐ molybdate   complex.
Reduction   of   this   complex   with   ascorbic   acid   produces   a   second   complex   with   a   deep   blue   color.
To   determine   the   concentration   of   the   complex   we   measure   its   absorbance   at   a   wavelength   of   880nm   by   using   an   instrument   called   a   spectrophotometer.
The   table   below   shows   the   transmittance   measured   for   a   series   of   solutions.
As   you   will   see   when   you   make   the   appropriate   plot,   something   went   wrong   with   one   of   the   standards.
You   will   therefore   make   calculation   in   two   different   ways:   with   all   the   data   and   with   the   outlaying   standard   excluded.
1
2
3
4
0.112
0.130
0.345
0.516
5   0.650
6
Detn1
1.272
Detn2
Detn3
0.971
0.883
0.853
0.727
0.606
0.454
0.313
0.435
0.457
0.423
Assignment   –overview
1.
Enter   these   data   in   a   spreadsheet.
2.
Calculate   the   absorbance   for   each   standard,   each   sample,   and   the   blank.
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3.
Correct   the   measured   absorbance   of   each   standard   and   each   sample   for   the   blank.
4.
Prepare   a   plot   of   blank ‐ corrected   absorbance   (y ‐ axis)   vs.
concentration   (x ‐ axis).
Be   sure   to   label   the   axes   and   to   include   units   as   appropriate.
5.
Use   the   analysis   tool   “ Regression ”   available   on   Excel   or   another   spread   sheet   to   obtain   four   linear   best   fits   to   the   data:   (draw   the   best   straight   line,   which   is   called   trendline   in   Excel;   not   curved   or   connected   )   y=mx+b   with   the   problematic   standard;   y=mx+b   without   the   problematic   standard;   y=mx   with   the   problematic   standard;   y=mx   without   the   problematic   standard.
6.
Calculate   the   average   absorbance   for   the   unknown.
7.
From   the   best ‐ fit   slopes   and   intercepts,   and   from   the   absorbance   of   the   unknown,   obtain   four   estimates   of   the   concentration   of   PO
4
3 ‐
(aq)   in   the   unknown   solution.
8.
Decide   which   estimate   of   the   concentration   of   PO
4
3 ‐
(aq)   you   think   is   the   best.
Label   all   parts   of   your   work   clearly.
The   detailed   instructions   below   assume   that   you   will   use   Microsoft   Office   Excel   2007,   which   is   installed   on   most   public   computers   on   campus.
You   may   use   a   different   version   of   Excel   or   other   spreadsheet   programs   if   you   wish.
Virtually,   all   spreadsheet   programs   have   “Help”   function   (in
Excel,   press   “F1”   key   to   go   to   the   “Help”   menu),   which   you   may   find   useful.
As   in   all   computer   work,   remember   to   save   your   work   often.
At   the   end   of   this   handout,   you   will   find   a   spreadsheet   that   I   have   prepared,   but   with   incomplete   numerical   results.
You   may   wish   to   use   it   as   a   model.
However,   notice   that   the   data   is   not   the   same.
1) Enter   the   data
Launch   the   spreadsheet   program.
The   previous   prelaboratory   assignment   “Analysis   of   a   titration   curve”,   gives   some   basic   instructions   for   how   to   enter   data.
Be   sure   to   save   your   work
frequently   as   you   go   along.
2) Calculate   the   absorbances   from   the   transmittances
A   simple   formula   relates   absorbance   to   transmittance.
Suppose   the   first   transmittance   is   in   cell
D7   .
We   want   to   put   the   calculated   absorbance   in   E7,   which   is   to   the   right   of   D7.
To   do   this,
enter   in   cell   E7   the   text   exactly   (including   the   equal   sign)
= ‐ log(D7)
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Note   that   the   “D”   may   be   either   upper   or   lower   case.
Note   also   that   your   formula   should   probably   not   contain   any   spaces.
If   you   enter   a   space,   the   spreadsheet   will   think   you   are
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entering   text   rather   than   a   mathematical   formula,   although   some   spreadsheets   are   smarter   than   others   in   this   aspect.
Remember   to   control   significant   figure   in   your   output!
Instructions   for   how   to   do   so   are   given   in   the   previous   prelaboratory   assignment.
You   can   repeat   the   calculation   of   the   absorbance   from   transmittance   for   each   row   of   the   table.
Alternatively   and   more   quickly,   you   can   copy   your   formula   to   calculate   absorbances   for   the   other   cells   as   follows.
•
By   using   the   mouse   or   the   arrow   keys,   place   the   cursor   in   cell   E7   (or   whatever   cell   you   want   to   copy   from)   and   right   click.
•
Choose   the   command   “ Copy.
”   (Experts,   use   [control]+C)The   cell   you   have   chosen   to   copy   from   will   develop   a   special   border   to   show   that   it   is   ready   to   be   copied.
•
Move   the   cursor   to   the   destination   or   copy ‐ to   cell   and   click   the   left   mouse   button.
•
Hit   the   return/enter   key.
(Experts   will   know   that   you   can   copy   to   more   than   once   cell   at   a   time   by   selecting   a   block   of   destination   cells   and   that   you   can   use   [control]+V   to   paste)
•
Be   sure   to   check   to   see   that   the   results   look   reasonable.
The   copy   command   has   some   special   properties   that   can   sometimes   lead   you   astray.
You   can   examine   the   formula   of   the   cell   with   the   newly   pasted   information   by   placing   the   cursor   there   and   click   the   left   mouse   button.
3) Correct   absorbances   for   blank
The   idea   here   is   subtract   the   blank   absorbance   from   the   absorbances   for   the   standards   and   for   the   samples.
Suppose   the   first   absorbance   you   want   to   correct   is   in   cell   E7   and   suppose   the   blank   value   is   in   cell   E4.
The   cell   to   the   right   of   E7   will   be   F7.
To   calculate   the   corrected   absorbance   in   cell   F7,   enter   in   cell   F7   the   text
=E7 ‐ $E$4
Make   sure   that   your   result   has   the   right   number   of   significant   figures.
To   calculate   blank ‐
corrected   absorbances   for   other   solutions   by   copying   this   formula,   proceed   as   described   in   the   previous   section.
Note   that   by   placing   the   “$”   symbol   in   front   of   the   “E”   and   in   front   of   the   “4”,
the   Excel   will   know   not   to   alter   the   cell   name   E7   when   you   copy   the   formula.
3
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An   alternative   is   to   enter   the   formula
=E7 ‐ constant
Where   “constant”   is   the   actual   numerical   value   of   the   absorbance   measured   for   the   blank.
4) Plot   absorbance   versus   concentration
Prepare   the   scatter   plot   with   right   format.
Plot   the   blank ‐ corrected   absorbance   vs   concentration   for   the   six   standard   solutions.
To   prepare   a   plot,   use   mouse   to   select   part   of   the   table   containing   the   blank ‐ corrected   absorbance   and   concentration   for   the   six   standards.
GO   to   the   “Insert”   menu,   in   the   “Chart”   panel,   under   “Scatter”,   select   “Scatter   with   only   Markers” .
Double ‐ Click   the   graph,   which   gives   you   access   to   “Chart   Tool”.
From   the   “Chart   Layouts”,   select   the   layout   which   allows   you   to   edit   the   “Chart   Title”   and   “Axis   Title”.
Double ‐ Click   the   “Chart   Title”   on   the   graph   to   label   your   graph.
Double ‐ Click   the   “Axis   Title”   to   label   the   axes   with   correct   units.
5) Obtain   a   best   fit   to   the   data
We   want   to   fit   a   straight   line   to   our   plot   of   the   blank ‐ corrected   absorbance   vs   concentration.
Blank ‐ corrected   absorbance=m   *   concentration   +   b
That   is,   we   want   the   values   of   m   (the   slope)   and   b   (the   intercept)   in   the   equation   above.
The   mathematical   process   used   to   obtain   m   and   b   is   called   regression   analysis.
Many   hand   held   calculators   offer   this   capability,   and   you   may   wish   to   use   a   hand   held   calculator   to   check   your   work.
Excel   not   only   does   what   hands   calculators   can   do   but   also   report   a   much   more   extensive   set   of   statistics   and   other   kinds   of   information.
To   use   the   Excel,   you   will   have   to   tell   it   which   data   to   fit   and   whether   you   want   to   force   fit   through   the   origin   (that   is,   choose   a   fit   in   which   the
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constant   b   is   forced   to   be   zero).
You   may   also   want   to   tell   the   spreadsheet   program   where   to   put   the   result ‐  but   if   you   don’t,   it   will   choose   its   own   place.
To   obtain   the   parameters   of   a   best ‐
fit   using   Excel,   proceed   as   follows.
Step   A   –Call   up   the   regression   software
First,   make   sure   that   Excel   has   the   regression   software   available.
To   do   so,   click   on   the   “Data”   menu   and   look   to   see   if   “Data   Analysis”   is   on   it.
If   not,   click   the   “Office”   button.
Click   “Excel
Options”   on   the   bottom   to   open   the   “Excel   Options”   dialog.
From   the   left   panel,   select” Add ‐
Ins” ,   on   the   bottom   of   the   right   panel,   you   will   find   “Manage” .
Choose   “ Excel   Add ‐ ins”   for
“Manage”   and   click   “Go” .
An   “Add ‐ Ins”   window   will   show   up.
Check   the   “Analysis   ToolPak”   and   click   “OK” .
Now   you   should   be   able   to   find   the   “Data   Analysis”   tool   from   the   “Data”   menu.
Read   “Load   the   Analysis   ToolPak”   section   of   Excel   Help.
Use   keyword   “ToolPak”   in   “Help”   to   search.
From   the   “Data”   menu,   select   “Data   Analysis”.
In   the   dialog   box   that   pops   up,   scroll   down   to
“Regression”   and   highlight   it   by   positioning   the   cursor   on   it   and   clicking   with   the   left   mouse   button.
Click   OK.
A   dialog   box   labeled   “Regression”   should   appear.
Steps   B ‐ D   have   comments   on
how   to   fill   in   the   “Regression”   dialog   box.
StepB ‐  “Regression”   dialog   box:   Input   the   location   of   the   data
Input   the   Y ‐ range   (blank ‐ corrected   absorbance   in   this   case).
Suppose   your   blank ‐ corrected   absorbance   data   are   in   cell   F7:F12.
Enter   the   text   into   the   dialog   blank   labeled   “Input   Y   Range”:
$f$7:$f$12
Input   the   X ‐ range   analogously.
Experts,   note   that   you   can   also   use   the   mouse   to   select   the   data
ranges.
For   Regression   to   work   properly   all   the   x   data   and   all   the   y   data   must   be   contiguous ‐  that   is   ,   the   x   data   must   appear   in   a   column   with   no   blank   rows   separating   them,   as   must   the   y   data.
Therefore,   before   you   can   carry   out   the   regression   analysis   with   one   standard   eliminated,   you   will   have   to   copy   the   concentration   and   corrected   absorbance   data   to   a   new   location   on   the   sheet.
If   you   are   unfamiliar   with   Excel,   it   may   be   easier   simply   retype   the   data   in   a   new   location.
Step   C ‐  “Regression”   dialog   box   continued:   Do   you   want   to   force   the   line   through   the   origin   (X ‐
0,   Y=0)?
If   you   want   to   force   the   best ‐ fit   line   through   the   origin,   then   check   the   option   “Constant   is
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Zero”.
Otherwise,   leave   this   unchecked.
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Step   D ‐ “Regression” dialog   box,   concluded:   Where   to   put   the   output?
Try   experimenting   with   this   one.
Just   make   sure   that   the   output   from   this   regression   will   not   write   over   something   that   you   have   already   put   on   the   spreadsheet   and   want   to   keep.
To   avoid   headache,   save   your   file   before   you   do   the   regression.
Step   E ‐ Interpreting   the   output   from   “Regression”
As   you   will   see,   “Regression”   produces   a   great   deal   of   statistical   information.
At   this   time   we   are   concerned   with   only   two   results.
Search   for   the   section   of   the   output   that   looks   like   this:
Your   numbers   will   be   different.
What   you   want   are   the   coefficients:   The   “intercept”   (what   we   call   b),   and   the   “X   Variable   1”   (i.e.,   the   slope   of   the   best   fit   line,   which   we   have   called   m).
Step   F ‐ Another   way   to   get   best ‐ fit   line.
Another   way   to   do   the   regression   analysis   is   to   add   trendline   to   your   plot   made   in   4).
However,   doing   so,   you   will   not   be   able   to   get   the   detailed   statistical   information   as   in   the   table   above.
On   the   graph,   right ‐ click   a   marker   (one   data   point),   then   choose   “Add   Trendline…” .
From   the
“Format   Trendline”   dialog,   under   “Trendline   Operations” ,   choose   “Linear”   for   the
“Trend/Regression   Type” .
For   the   2   plots   in   which   you   want   to   force   the   trendline   to   go   through   the   origin,   check   “Set   Intercept”   and   set   the   value   to   “0.0”.
Leave   this   unchecked   for   the   other   two   plots.
Check   “Display   Equation   on   Chart”   to   have   the   equation   representing   the   trendline   on   the   chart.
The   equation   will   have   format   y=mx+b   or   y=mx,   depends   on   whether   you   force   the   trendline   to   go   through   the   origin   or   not.
The   value   of   slop   m   and   intercept   b   can   be   read   from   equations   on   the   charts.
From   your   graph,   you   can   always   right ‐ click   on   the   trendline   to   access   the   “Format   Trendline”   dialog.
Read   “Add,Change   or   remove   a   trendline   in   a   chart”   section   for   detailed   instruction   from   the
Excel   “Help”.
Search   “Trendline”   in   help   will   lead   you   to   this   section.
6) Calculate   the   average,   blank ‐ corrected   absorbance   of   the   unknown
Suppose   the   three   blank ‐ corrected   absorbance   values   for   the   unknown   are   in   cells   F15:F17.
To   calculate   the   average   blank ‐ corrected   absorbance   in   cell   F18,   enter   in   cell   F18   the   text
=average(F15:F17)
6
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For   further   reference,   you   may   be   interested   that   the   text =stdev(F15:F17)   returns   the   standard   deviation   of   the   data   set   and   that   sure   beats   calculating   the   standard   deviation   by   hand.
7) Calculate   the   concentration   of   the   unknown
Your   best   fits   should   be   of   two   forms:
Blank ‐ corrected   absorbance   =m   *Concentration   (ppm)   +   b
Here   m   is   the   “X   Variable”   and   b   is   the   “Intercept”   from   above   (what   are   the   units   of   m   and   b?).
and
Blank ‐ corrected   absorbance=m*   Concentration   (ppm)
The   second   form   is   what   you   assume   when   you   “force   the   fit   through   the   origin” .
In   this   case,   only   the   value   if   the   slope,   m,   is   free   to   vary.
Rearrange   these   formulas   to   express   concentration   in   terms   of   blank ‐ corrected   absorbance,   the
slope   m,   and,   when   appropriate,   the   intercept   b.
Enter   these   rearranged   formulas   in   different   cells,   beginning   with   the   “=”   character   and   use   them   to   calculate   the   concentration   of   phosphate   ion   in   unknown.
A   model   solution   is   attached.
Blanked ‐ out   areas   appear   where   you   should   make   entries.
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