Statistical evaluation
Other answers have done a good job of explaining the chemistry. I wanted to add a purely statistical explanation of why your comparison is misleading.
It is misleading to compare fits solely by the error percentage. The reason is that each model uses a different number of parameters.
Exponential fit
$$y = a \exp(b x)$$
This has two parameters, $a$ and $b$.
Quadratic fit
$$y = a x^2 + bx + c$$
This has three parameters, $a$, $b$, and $c$.
Cubic fit
$$y = a x^3 + bx^2 + cx + d$$
This has four parameters, $a$, $b$, $c$, and $d$.
Quartic fit
$$y = a x^4 + bx^3 + cx^2 + dx + f$$
This has five parameters, $a$, $b$, $c$, $d$, and $f$.
What happens as you add more parameters
John von Neumann, a well-known mathematician, famously said:
With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.
Adding more parameters will always make a fit better, whether or not the parameters are physically meaningful.
An information criterion
Statisticians have devised various metrics to compare models with different numbers of parameters. Some of them are the Aikake information criterion and the Bayesian information criterion. The core idea of these metrics is the same -- penalize models with more parameters. If a simple model with two parameters fits the data almost as well as a complicated one with five parameters, the simple model is likely a much better description of the data.
Those two information criteria are evaluated from the likelihood, a fancy statistical number which turns out to be closely related to the error that you have already calculated. Let's take the example of the AIC:
$$AIC = 2k - 2\ln{L}$$
Here, $k$ the number of parameters and $L$ is the likelihood. Lower AIC scores are better. Unlike just comparing relative error, comparing AICs between models with different numbers of parameters is OK, because the AIC takes the extra parameters into account.
It's easy to compute these information criteria using modern statistical packages like R. For example, here's some R code:
require(tidyverse)
require(broom)
conc <- c(0.2, 0.153, 0.124, 0.104, 0.09, 0.079, 0.070, 0.063, 0.058, 0.053, 0.049)
time <- c(0, 20, 40, 60, 80, 100, 120, 140, 160, 180, 200)
df <- tibble(conc=conc, time=time)
model_list <- list('quadratic' = conc ~ poly(time, 2),
'cubic' = conc ~ poly(time, 3),
'quartic' = conc ~ poly(time, 4),
'exponential' = log(conc) ~ time,
'second_order' = I(1 / conc) ~ time)
num_pts <- length(conc)
num_models <- length(model_list)
fit.results <-
df %>%
map(model_list, lm, .) %>%
map_df(glance)
fit.results
That gives the following result:
$$
\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline
model & r.squared & adj.r.squared & sigma & statistic & p.value & df & logLik & AIC & BIC & deviance & df.residual \\ \hline
quadratic & 0.9813919 & 0.9767399 & 0.007232647 & 210.9601 & 1.198971e-07 & 3 & 40.36382 & -72.72765 & -71.13607 & 4.184895e-04 & 8 \\ \hline
cubic & 0.9977167 & 0.9967381 & 0.002708474 & 1019.5748 & 1.323163e-09 & 4 & 51.90266 & -93.80533 & -91.81585 & 5.135082e-05 & 7 \\ \hline
quartic & 0.9997019 & 0.9995032 & 0.001057037 & 5030.5170 & 1.059279e-10 & 5 & 63.10056 & -114.20113 & -111.81376 & 6.703963e-06 & 6 \\ \hline
first-order (exponential) & 0.9634713 & 0.9594125 & 0.091628408 & 237.3814 & 8.937773e-08 & 2 & 11.78552 & -17.57104 & -16.37735 & 7.556189e-02 & 9 \\ \hline
second-order & 0.9999197 & 0.9999107 & 0.048309187 & 112032.5265 & 9.653383e-20 & 2 & 18.82683 & -31.65367 & -30.45998 & 2.100400e-02 & 9 \\ \hline
\end{array}
$$
This result shows you the AIC (and BIC) for each model. You are right that the higher-order models fit better, but not by as much as you would think from looking at the relative error. The quartic model as an AIC of -114, and the quadratic model has an AIC of -72. So the quartic model is better (in a statistical, not necessarily chemical) sense, but only by twofold. You have to penalize the extra parameters you gave it.
Chemical evaluation
The only two models which make chemical sense are the last two: an exponential fit (i.e. first-order reaction) and a inverse fit (i.e. second-order reaction). Between those two, the second-order fit is much better than the exponential, with an AIC of -31 compared to -17 for the exponential fit.
Graph
There's no substitute for graphing your fits.
fit.data <- df %>%
map(model_list, lm, .) %>%
map_df(~augment_columns(data=df, x = .)) %>%
mutate(model = rep(names(model_list), each = num_pts)) %>%
mutate(.fitted = ifelse(model == 'second_order', 1/.fitted, .fitted)) %>%
mutate(.fitted = ifelse(model == 'exponential', exp(.fitted), .fitted))
options(repr.plot.height=4, repr.plot.width=5)
fit.data %>%
ggplot(aes(x=time, y=conc)) +
geom_point() +
geom_line(aes(x=time, y=.fitted, color=model)) +
theme_bw()
ggsave('SEChem_65921.png', height=4, width=5)
Results in:
It shows the deficiency of the first-order (exponential) fit, as well as the deficiency with the quadratic fit. It shows that the other models fit pretty well, but keep in mind that only some of these make chemical sense.
Conclusion
Curve-fitting and parameter estimation to fit chemical reaction data requires both statistical and chemical expertise. You have to (a) make sure you don't give too much credit models with lots of parameters, and (b) make sure your fits make chemical sense. For your data, a second-order fit seems best, because it (a) gives a good AIC, (b) has few parameters, and (c) makes chemical sense.
