Phenol red is an indicator commonly used to measure pH in swimmingpool test kits, see e.g.[2]. The goal ofthiscolorSpec vignette is to reproduce the colors seenin such a test kit, for typical values of pool pH. Calculations likethis one might make a good project for a college freshman chemistryclass. Featured functions in this vignette are:interpolate() andcalibrate().
Absorbance Spectra at Different pH Values
The absorbance data for phenol red has already been digitized from[1]:
path=system.file("extdata/stains/PhenolRed-Fig7.txt",package="colorSpec" )wave=350:650phenolred=readSpectra( path,wavelength=wave )par(omi=c(0,0,0,0),mai=c(0.6,0.7,0.4,0.2) )plot( phenolred,main='Absorbance Spectra of Phenol Red at Different pH Values' )
Compare this plot with[1], Fig. 7.Unfortunately, the concentration and optical path length are unknown,but these curves can still be used as ‘relative absorbance’.
Absorbance at Selected Wavelengths
We investigate how absorbance depends on pH for a few selectedwavelengths.
wavesel=c(365,430,477,520,560,590 )# 365 and 477 are 'isosbestic points'mat=apply(as.matrix(wavesel),1,function( lambda ) {as.numeric(lambda== wave) } )colnames( mat )=sprintf("%g nm", wavesel )mono=colorSpec( mat,wavelength=wave,quantity='power' )RGB=product( mono, BT.709.RGB,wavelength=wave )# this is *linear* RGBcolvec= grDevices::rgb(SignalRGBfromLinearRGB( RGB/max(RGB),which='scene' )$RGB )phenolsel=resample( phenolred, wavesel )pH=as.numeric(sub('[^0-9]*([0-9]+)$','\\1',specnames(phenolred) ) )pHvec=seq(min(pH),max(pH),by=0.05)phenolsel=interpolate( phenolsel, pH, pHvec )mat=t(as.matrix( phenolsel ) )par(omi=c(0,0,0,0),mai=c(0.8,0.9,0.6,0.4) )plot(range(pH),range(mat),las=1,xlab='pH',ylab='absorbance',type='n' )grid(lty=1 ) ;abline(h=0 )matlines( pHvec, mat,lwd=3,col=colvec,lty=1 )title("Absorbance of Phenol Red at Selected Wavelengths")legend('topleft',specnames(mono),col=colvec,lty=1,lwd=3,bty='n' )
Note that the curves for the isosbestic points 365 and 477 nm areapproximately flat, as expected. But for 430 nm the curve is distinctlynon-monotone. This indicates that the solution is not truly a mixture ofthe acidic and basic species (especially for pH\(\le\) 6), and there may be an undesiredside reaction, see[3].
Interpolation from pH=6.8 to pH=8.2
Swimming pools should be slightly basic; a standard test kit coversthe range from pH=6.8 to pH=8.2.
pHvec=seq(6.8,8.2,by=0.2)phenolpool=interpolate( phenolred, pH, pHvec )par(omi=c(0,0,0,0),mai=c(0.6,0.7,0.4,0.2) )plot( phenolpool,main="Absorbance Spectra of Phenol Red at Swimming Pool pH Values" )
The rest of this section is best viewed on a display calibrated forsRGB, see[4].
# create an uncalibrated 'material responder'testkit=product( D65.1nm,'solution', BT.709.RGB,wave=wave )# now calibrate so that fully transparent pure water has response RGB=c(1,1,1)testkit=calibrate( testkit,response=1 )RGB=product( phenolpool, testkit )RGB
## R G B## pH=6.8 1.0282473 0.6840105 0.2260205## pH=7 1.0237233 0.5938036 0.2506686## pH=7.2 1.0182971 0.4961330 0.2788955## pH=7.4 1.0124869 0.4022827 0.3100400## pH=7.6 1.0067433 0.3195781 0.3440245## pH=7.8 1.0014227 0.2505541 0.3812950## pH=8 0.9969035 0.1947707 0.4225796## pH=8.2 0.9935212 0.1503867 0.4683717
Unfortunately, in some cases the red value is greater than 1 (G and Bare OK). The color is outside the sRGB gamut. Start over andrecalibrate.
testkit=product( D65.1nm,'solution', BT.709.RGB,wave=wave )# recalibrate, but lower the background a little, to allow more 'headroom' for indicator colorsbglin=0.96# graylevel for the background, lineartestkit=calibrate( testkit,response=bglin )RGB=product( phenolpool, testkit )# this is *linear* sRGBRGB
## R G B## pH=6.8 0.9871174 0.6566501 0.2169797## pH=7 0.9827743 0.5700514 0.2406419## pH=7.2 0.9775652 0.4762877 0.2677396## pH=7.4 0.9719874 0.3861913 0.2976384## pH=7.6 0.9664736 0.3067950 0.3302635## pH=7.8 0.9613658 0.2405320 0.3660432## pH=8 0.9570273 0.1869799 0.4056764## pH=8.2 0.9537803 0.1443713 0.4496368
All values have been multiplied bybglin, and are nowOK. Draw the RGB patches on a white background multiplied by the sameamount.
df.RGB=data.frame(LEFT=1:nrow(RGB),TOP=0,WIDTH=1,HEIGHT=2 )df.RGB$RGB= RGBpar(omi=c(0,0,0,0),mai=c(0.3,0,0.3,0) )plotPatchesRGB( df.RGB,space='sRGB',which='scene',labels=F,background=bglin )text( (1:nrow(RGB))+0.5,2,sprintf("%.1f",pHvec),adj=c(0.5,1.2),xpd=NA )title(main='Calculated Colors for pH from 6.8 to 8.2' )
The background color is that of pure water, and is not the fullRGB=(255,255,255).
In the first figure above, the phenol red concentration and opticalpath length are unknown. Compared to a real test kit, the calculatedcolors look a little faded. An absorbance multiplier can easily tweakthe unknown concentration, as follows.
tweak=1.3phenolpool=multiply( phenolpool, tweak )df.RGB=data.frame(LEFT=1:nrow(RGB),TOP=0,WIDTH=1,HEIGHT=2 )df.RGB$RGB=product( phenolpool, testkit )# this is *linear scene* sRGBpar(omi=c(0,0,0,0),mai=c(0.3,0,0.3,0) )plotPatchesRGB( df.RGB,space='sRGB',which='scene',background=bglin,labels=F )text( (1:nrow(RGB))+0.5,2,sprintf("%.1f",pHvec),adj=c(0.5,1.2),xpd=NA )main=sprintf('Calculated Colors for pH from 6.8 to 8.2 (absorbance multiplier=%g)', tweak )title(main=main )
These colors are a better match to those in the test kit.