Alcohol vaporization by 5%, How much liters remained in the bottle?

Question

Problem:

"There is a 30% solution of alcohol in water in a bottle of 6 liters. Some of the solution is vaporized, with twice (2x) as much alcohol vaporized than water. After the vaporization the remaining solution is 25% alcohol.

What is the total amount of new amount of liters inside the bottle after the vaporization occurred?

Solution attempt:

$alcohol=2*water \space \to vaporized$

initial values: $$alcohol \to \space 0.3*6l=1.8l$$

$$water\space\space \to \space 0.7*6l=4.2l$$ $$alcohol=water*2; \space $$ $$------------------------------$$

$$1.8-(0.25*1.8)=1.8-0.45=1.35l \space\space alcohol \space left$$ $$0.45/2=0.225l \space\space water\space reduction$$ $$amount_{new}=1.35+ (4.2-0.225)=5.32l$$

The result should be 5.28.

What did I miss?

Answer 1

If you have six unknowns $a_\text{start},w_\text{start},a_\text{ev},w_\text{ev},a_\text{left},w_\text{left}$ and six equations: $$a_\text{start}+w_\text{start}=6$$ $$a_\text{start}=0.3 (a_\text{start}+w_\text{start})$$ $$a_\text{ev}=2w_\text{ev}$$ $$a_\text{left}=0.25 (a_\text{left}+w_\text{left})$$ $$a_\text{ev}=a_\text{start}-a_\text{left}$$ $$w_\text{ev}=w_\text{start}-a_\text{left}$$ then you can solve these to give $$a_\text{start}=1.8$$ $$w_\text{start}=4.2$$ $$w_\text{left}=3a_\text{left}$$ $$(1.8-a_\text{left})=2(4.2-3a_\text{left})$$ $$a_\text{left} = 1.32$$ $$w_\text{left} = 3.96$$ $$a_\text{ev} = 0.48$$ $$w_\text{ev} = 0.24$$

and that gives the total amount left as $a_\text{left}+w_\text{left}=5.28$

A single calculation just using the numbers in the question could be $\dfrac{ 2 \times 6 \times (1-0.3) - 6 \times 0.3}{1 +0.25}=5.28$ but I think that would be less clear

Answer 2

Originally, the solution contained $1.8l$ alcohol and $4.2l$ water.

Let the volume of water that evaporates be $x$, then the vol. of alcohol that evaporates is $2x$.

The final alc. concentration is $25 \%$ ($1$ in $4$). That means the final proportion of alc. to water is $1:3$ ($1$ is to $3$).

So $3(1.8 - 2x) = 4.2 - x$

$x = 0.24$

Hence the total volume that evaporates is $3x = 0.72l$, giving a final volume of $6- 0.72 = 5.28l$.

Answer 3

$$\begin{array} & \hline &\text{Water}&\text{Alcohol} &\text{Total}\\ \hline \;\; \text{Original}&4.2 &1.8 &\color{blue}{6.0}\\ &(70\%)&(\color{blue}{30\%})&(100\%)\\\\ -\text{Evaporated}&-V &-\color{blue}{2}V &-3V\\\\ \hline\\ \;\; \text{Balance (Final)} &4.2-V &1.8-2V &\color{green}{6-3V}\\ &(75\%) &(\color{blue}{25\%}) &(100\%)\\ \hline \end{array}$$ $$6-3V=4(1.8-2V)\\ \Rightarrow V=0.24$$ Hence final total volume of $\color{red}{5.28\ell}$.

---

NB: Colour code as follows:

$\color{blue}{\text{Blue}:}$ Given
$\color{green}{\text{Green}:}$To be determined