Question

If you have ever swum in a pool and your eyes began to sting and turn red, you are aware of the effects on an incorrect pH level. The pH level measures the concentration of hydronium ions and can be modeled by the function p(t) = –log10 t. The variable t represents the amount of hydronium ions, and p(t) gives the resulting pH level.

Water at 25 degrees Celsius has a pH of 7. Anything that has a pH lower than 7 is called acidic, while a pH above 7 is basic, or alkaline. Seawater has a pH just over 8, while lemonade has a pH of approximately 3.

1. Create a graph of the pH function. Locate on your graph where the pH value is 0 and where it is 1. You may need to zoom in on your graph.
2. A pool company forgets to bring their logarithmic charts, but they need to raise the amount of hydronium ions in a pool by 0.50. Using complete sentences, explain how your graph can be used to solve 10–y = 0.50. Find the approximate solution.
3. The pool company has developed new chemicals that transform the pH scale. Using the pH function p(t) = –log10 t as the parent function, explain which transformation would result in a y-intercept. Use complete sentences and show all translations on your graph.
   
   • p(t) + 1
   • p(t + 1)
   • –1 • p(t)

Answer 1

1. The graph of p(t)=-log(10t) is zero when 10t=1, that is, when t=0.1. The p(t) axis is an asymptote since at t=0, -log(10t) approaches infinity. Therefore the graph starts close to the asymptote when t is just greater than zero, drops very steeply, and crosses the t axis at t=0.1, then it becomes negative but the curve is much shallower. At t=1, p(1)=-1, and p(10)=-2, so the graph drops only 1 unit negatively while t goes from 1 to 10. The scale of both axes needs to be generous, so as to distinguish between values only tiny fractions apart. The pH value of 7 corresponds to a value of 10t=10^-7 or t=10^-8. A pH of 3 corresponds to t=10^-4. For pH values of a reasonable range it is probably best to graph between 0 and 1 given the range 0 to infinity of the pH values, while t goes from 0 to 0.1. If 10-y=0.50 then y=9.50 and -log(10t)=9.50, so 10t=10^-9.50=3.1623*10^-10 and t=3.1623*10^-11.
2. p(t)=-log(10t) so p(1.5t)=-log(15t), where t has increased by 0.50 of its value. The difference between the pH values is p(1.5t)-p(t)=-log(15t)+log(10t)=log(10/15)=log(2/3)=-0.1761, so the pH value falls (becomes more acidic); and p(t+0.5)=-log(10t+5), which requires a specific value for t. p(t+0.5)-p(t)=-log(10t+5)+log(10t)=-log(1+0.5/t). If t is small, as it usually is, 1/t is large and this becomes -log(0.5/t) or log(t/0.5)=log(2t) or log(t)+0.3010. Log(t) is negative so the result is more positive or alkaline. If y=p(t) and 10-y=0.50, y=9.50 and 10t=3.162*10^-10, making t=3.162*10^-11. This is the value of t for a pH of 9.50.
3. p(t+1) would shift the graph 1 unit to the left, so that when t=0 p=-1, making the y-intercept -1.

 

Comments

The function for pH is ambiguous in your question. Is it log(10t) or log[10](t)? I didn't draw the graph, partly because of this ambiguity. Log(10t) means to me log to the base 10 of 10t and log[10](t) means log to the base 10 of t. If you can clarify which meaning is intended (I think it's probably P(t)=-log to the base 10 of t) then I can attempt to show you the graph. Chemistry or physics textbooks use log to the base 10 of the hydrogen ion concentration. Here's one graph with zoom in:

A pH of 1 corresponds to a H+ concentration of 0.1. A pH value of 0 corresponds to H+ of 1. The subdivisions of the horizontal scale are 0.05 steps. Two steps=0.1, with a pH of 1.

Ignore my original answers. I can come back to the other parts of your question in due course.

If the H+ concentration is increased by 0.50, then 0.5 becomes 0.75, 1 becomes 1.5, etc. So instead of 0.25, 0.5, 0.75 and 1 read off 0.375, 0.75, 1.125 and 1.5. For example, the pH for H+=.50 is about 0.3; if H+ is increased to 0.75, the pH drops to about 0.125.

If p(t)=-log(10t), then this is the same as p(t)=-(1+log(t))=-1-log(t) or p(t)+1=-log(t). This is a vertical displacement of the graph:

In this graph, the yellow curve corresponds to y=-1-log(t) and the blue line shows where the t-axis is displaced downwards by 1.

 

The horizontal tan line near the top of the graph is at pH=1 in this magnification. On the red curve t=0.1 but on the displaced yellow curve t=0.01 (10t=0.1).

10-y=0.50, so y=9.5 represents a pH value of 9.5 (green line):

This means that the concentration of H+ is very low because the curve is almost touching the pH axis. For 9.50=-log(t), log(t)=-9.50=-10+0.50 so t=3.162*10^-10; for 9.50=-log(10t), t=3.162*10^-11.

p(t+1)=-log(10t+10); when t=0 this becomes -log(10)=-1. The yellow curve in graph 2 is displaced 1 to the left so that it cuts the pH axis at -1, the y-intercept.