Phenol red is an indicator commonly used to measure pH in swimming
pool test kits, see e.g. [2]. The goal of
this colorSpec vignette is to reproduce the colors seen
in such a test kit, for typical values of pool pH. Calculations like
this one might make a good project for a college freshman chemistry
class. Featured functions in this vignette are:
interpolate() and calibrate().
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 = 350:650
phenolred = 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 selected
wavelengths.
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* RGB
colvec = 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 are
approximately flat, as expected. But for 430 nm the curve is distinctly
non-monotone. This indicates that the solution is not truly a mixture of
the acidic and basic species (especially for pH \(\le\) 6), and there may be an undesired
side reaction, see [3].
Interpolation from pH=6.8 to pH=8.2
Swimming pools should be slightly basic; a standard test kit covers
the 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 for
sRGB, 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 B
are OK). The color is outside the sRGB gamut. Start over and
recalibrate.
testkit = product( D65.1nm, 'solution', BT.709.RGB, wave=wave )
# recalibrate, but lower the background a little, to allow more 'headroom' for indicator colors
bglin = 0.96 # graylevel for the background, linear
testkit = calibrate( testkit, response=bglin )
RGB = product( phenolpool, testkit ) # this is *linear* sRGB
RGB
## 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 by bglin, and are now
OK. Draw the RGB patches on a white background multiplied by the same
amount.
df.RGB = data.frame( LEFT=1:nrow(RGB), TOP=0, WIDTH=1, HEIGHT=2 )
df.RGB$RGB = RGB
par( omi=c(0,0,0,0), mai=c(0.3,0,0.3,0) )
plotPatchesRGB( df.RGB, space='sRGB', which='scene', labels=F, background=bglin )
text( (1:nrow(RGB)) + 0.5, 2, sprintf("%.1f",pHvec), adj=c(0.5,1.2), xpd=NA )
title( main='Calculated Colors for pH from 6.8 to 8.2' )
The background color is that of pure water, and is not the full
RGB=(255,255,255).
In the first figure above, the phenol red concentration and optical
path length are unknown. Compared to a real test kit, the calculated
colors look a little faded. An absorbance multiplier can easily tweak
the unknown concentration, as follows.
tweak = 1.3
phenolpool = multiply( phenolpool, tweak )
df.RGB = data.frame( LEFT=1:nrow(RGB), TOP=0, WIDTH=1, HEIGHT=2 )
df.RGB$RGB = product( phenolpool, testkit ) # this is *linear scene* sRGB
par( 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.