Phenol Red - pH Indicator"
In colorSpec: Color Calculations with Emphasis on Spectral Data

Phenol red is an indicator commonly used to measure pH in swimming pool test kits, see e.g. @K-1000. 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().

library( colorSpec )
library( spacesRGB )    # for functions plotPatchesRGB() and SignalRGBfromLinearRGB()

Absorbance Spectra at Different pH Values

The absorbance data for phenol red has already been digitized from @Rovati:

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 @Rovati, 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 @wiki:pH.

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 @wiki:sRGB.

# 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

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

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 )