Chapter 10.7, Problem 16P

To determine

To determine how longer it would take until 90% of the sugar is dissolved.

Answer to Problem 16P

After approximately 6.64 minutes the sugar will be 90% dissolved.

Explanation of Solution

Given information:

After 2 minutes 50% of the sugar is dissolved.

We shall use the formula for half-life,

  $N=N_0\cdot {\left(\frac{1}{2}\right)}^{\frac{t}{h}}$

(If an amount $N_0$ has a half-life $h$ , then the remaining substance in time $t$ is given by this formula)

Hence we put the values given in the question

After 2 minutes, 50% of sugar dissolved.

Therefore,

  $50=100\cdot {\left(\frac{1}{2}\right)}^{\frac{2}{d}}$ (Initial quantity is 100% and as 50% is mixed hence 50% is remaining)

Hence,

  $\begin{array}{c}50=100\cdot {\left(\frac{1}{2}\right)}^{\frac{2}{d}}\\ \frac{1}{2}={\frac{1}{2}}^{\frac{2}{d}}\\ \frac{2}{d}=1\\ d=2\end{array}$

Hence we got the value of $d=2$

Now time until 90% sugar dissolve is,

  $\begin{array}{l}10=100\cdot {\left(\frac{1}{2}\right)}^{\frac{t}{2}}\\ \frac{1}{10}={\frac{1}{2}}^{\frac{t}{2}}\end{array}$

Now we solve by taking logarithm both sides we get,

  $\begin{array}{l}\log 0.1=\frac{t}{2}\log 0.5\\ t=\frac{2\log 0.1}{\log 0.5}\\ t=6.64\end{array}$

Hence after approximately 6.64 minutes the sugar will be 90% dissolved.