Online Test Scenario:
A chemical reaction using a fixed amount of a compound produces heat and this is
measured in $kCal$. When a catalyst is added to the compound the amount of heat
produced increases. The increase in heat $(h)$ is believed to be approximately
proportional to the square root of the mass of catalyst added $(c)$.
Question:
Which one of the following plots is most likely to show a linear relationship between the variables?
- $\sqrt{h}$ against $c$
- $h$ against $c^{2}$
- $h$ against $\log{}{c}$
- $\log{}{h}$ against $\log{}{c}$
My attempt:
I've never come across a scenario in my studies like this (except for my masters thesis in altering axis to give a better line of best fit). Based on that logic and the wording "increases in $h$ is proportional to the $\sqrt{c}$"
i would say the correct answer is either $1)$ or $2)$ but i don't know which or the explanation for it.
Best Answer
The words
expressed algebraically say that there is some constant $k$ such that $$ h \approx k \sqrt{c}. $$ That is clearly not a linear relationship. Squaring doesn't help, because $$ h^2 \approx k^2c $$ isn't linear either.
Logarithms will deal with the square root: $$ \log h \approx \log k + (1/2)\log c . $$ That is linear.