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BY-NC-ND 4.0 license Open Access Published by De Gruyter August 4, 2022

Difficulties of novice students in solving the final concentration value of a mixture of solutions

  • Elvira Vaccaro , Carlos Stella and Manuel Alonso ORCID logo EMAIL logo


In this paper, we investigated and classified the answers of college freshmen when asked about “the final concentration value of a mixture of solutions ”. Prior to the explanation of the topic in class, a diagnostic questionnaire on “solutions” was presented to 532 first year students in the chemistry course at the University of Buenos Aires. The questionnaire consisted of three questions assessing the same concept: the calculation of the final concentration of a solution obtained mixing a concentrated and a dilute solution of the same solute. The format of the three questions was multiple choice answer with justification, but they differed in their chemical language style: chemical formulas, verbal-procedural, and visual languages were used. It was noted a trend to apply mathematical calculations, when chemical problems are addressed, even when such calculations are not necessary. Thus, obtaining a numerical result would be considered appropriate by the students, with no analysis of the significance of the value obtained. Nevertheless, question which uses visual language was answered correctly by a greater number of students. This would allow inferring that the use of this language brings students closer to a better understanding of the situation.


The concept of chemical solutions and their properties, as well as the resolution of exercises and problems that involve calculating the concentration of solutions, constitute topics of great interest in the teaching and learning of Chemistry, both in high school (Çalýk & Ayas, 2005; Nappa, Insausti & Sigüenza, 2005; Pinarbasi & Canpolat, 2003), and in college (Bekerman, Pepa, Vaccaro, Alonso, & Galagovsky, 2016; Chong, 2016; de Berg, 2012; Jadhav, Nair, Rayate, & More, 2019; Raviolo & Farré, 2018, 2020a, 2020b). The knowledge and understanding of these issues, beyond their intrinsic conceptual value, are necessary to address other topics within the discipline (Özden, 2009), such as chemical equilibrium, acid-base equilibrium, and chemical kinetics. In addition, solutions are the medium where many chemical processes take place (Atkins & Jones, 2005, pp 408).

However, numerous studies show the difficulties that students have about understanding and solving exercises and problems about concentration of solutions (Çalýk, Ayas, & Ebenezer, 2005; Özden, 2009). Such difficulties also include the resolution of those situations in which mixing of solutions is carried out and it is requested to determine the final concentration. In the latter case, our experience in the first year of the University shows that novice students add the concentrations of the solutions or confuse different expressions of concentration. These erroneous answers could have been originated, in part, in the way the questions are formulated (Danili & Reid, 2005; Talanquer, 2006). In this sense, Chemistry uses different types of language such as mathematical and chemical formulas, verbal-procedural language, visual language, and graphic language (Galagovsky & Bekerman, 2009; Lemke, 1998; Taber, 2015). These languages associate with representations of the concepts at the macroscopic, sub-microscopic and symbolic levels proposed by Johnstone (1982, 1991 for the teaching of natural sciences and especially for the teaching and didactic research of chemistry (Galagovsky, Rodríguez, Stamati, & Morales, 2003; Taber, 2013).

According to these authors, the language of mathematical and chemical formulas allows to formalize a chemical concept through symbols and procedures of these disciplines (relationships between variables, use of algorithms, equations and chemical symbols, etc.), and therefore corresponds to a symbolic level of representation. On the other hand, the visual language utilizes images, and thus, it appeals predominantly to the senses and, therefore, at a macroscopic level of representations. It should be noted that the representations of atoms and molecules, and graphics (e.g., coordinate graphs, statistical graphs), although use images, refer to the sub-microscopic and symbolic levels, respectively, and not to the macroscopic level (Galagovsky et al., 2003). Finally, and always with reference to the authors cited in the preceding paragraph, verbal-procedural language appeals to the description of laboratory operations necessary to carry out an experiment, and generally employ words specific to the discipline. This language presents, therefore, symbolic and macroscopic level contents. These three representation levels, to which the different languages can be associated, are related and used in a combined and integrated way for the understanding of concepts (Sirhan, 2007), and presentation and resolution of problems (Lemke, 1998).

This diversity and combination of languages could be, at the least, one of the difficulties that cause these wrong answers of students when answering exercises and problems related to mixing solutions.

Taking into account these considerations, in this paper, we investigate, in college freshmen, the influence of these languages on problem solving final concentration of solutions, obtained mixing a concentrated and a dilute solution of the same solute. For this purpose, we used questions in which each of the different languages of Chemistry prevails.


The questionnaire

Before the explanation of the topic in class, a diagnostic questionnaire on “solutions” was presented to 532 first year students in the chemistry course at the University of Buenos Aires.

The questionnaire consisted of three questions (Box 1), assessing the same concept: the calculation of the final concentration of a solution obtained from mixing a concentrated and a diluted solution of the same solute. The format of the three questions was multiple choice, but it was requested to provide a short explanation of the answer; and they were formulated using three different types of chemical languages previously mentioned in the introduction: mathematical and chemical formulas, verbal-procedural language, and visual language.

Box 1:

Diagnostic questionnaire taken to 532 student volunteers on the topic “concentration of solutions.”

Question 1: A chemist mixed two solutions of Na2CO3, whose concentrations were 0.1 g/L and 0.5 g/L respectively. The concentration of the resulting solution may be:
(a) 0.1 g/L
(b) 0.6 g/L
(c) 0.3 g/L
(d) 0.001 g/L
Choose an option and justify briefly

Question 2: A chemist has two beakers (“A” and “B”) with the same volume of two colorless liquids. He pours both of them into a jug “C”. The beaker “A” contained 100 g of sugar per liter of water, and the beaker “B” had 500 g of sugar per liter of water. What will be the concentration of the resulting solution in the jug “C”? Choose the correct option and justify briefly.
(a) 100 g/L
(b) 600 g/L
(c) 300 g/L
(d) 10 g/L
(e) None of the above answers is right.

Question 3: Two solutions were prepared in two beakers (B and D) containing the same amount of water. In the beaker B, 1 g of a purple solid substance (potassium permanganate, KMnO4) was dissolved. In the beaker D, 6 g of the same substance were dissolved. In the following illustration, the appearance of the two solutions is shown.
If now equal parts of B and D solutions are mixed in another container, what color do you think will be the resulting solution? Choose an option and justify.
Note: The colors of the images of the B and D solutions of both figures are the same.

So, the question (1) described the solute by a chemical formula and provided concentrations in grams per liter (g/L), therefore it resorted to a language of formulas. Question (2) reported experimental procedures; hence it used a verbal-procedural language. Finally, question (3) described the solute by a chemical formula, indicating that it was a purple-colored substance. In this case, different shades were used to illustrate concentrations, thus it resorted to macroscopic visual language. Although the students had learned the topic “concentration of solutions” in their pre-university studies, we thought that we could detect differences in the answers due to the type of chemical language used in each question. In the three questions, the concentration of the solution is expressed in units of mass of the compound per volume of the solution, a mathematical formalization of a chemical concept. However, in questions 2 and 3, this formalization is not necessary to arrive at the answer. In fact, question 2 also contains procedural language and question 3 visual language. In other words, these statements present more data than required, with the intention of investigating whether the students manage to arrive at a conceptually correct answer without the need to use mathematical and formula language, or, at least, if they verify whether the numerical answer obtained is conceptually correct.

Expected answers

The optional answers for all questions followed the same pattern:

Option (a): concentration identical to the more dilute solution in the problem (incorrect). Choosing this option would not take into account the influence of adding a second solution of a different concentration.

Option (b): concentration greater than the most concentrated solution in the problem (incorrect). This answer implied the sum of the concentration values, which would suggest that students regarded concentration as an extensive property.

Option (c): an intermediate concentration (correct).

Option (d): concentration lower than that of the most dilute solution concentration (incorrect). This option is absurd as the final concentration is lower than either of the two solution concentrations involved. Only students without any knowledge or intuition about the subject, or who reply at random could choose this answer.

Basics on using various chemical languages

Question 1: this question uses mathematical and chemical formulas language. It is similar to the questions which would be used in examinations at the end of the course. To resolve this problem, students should have acquired prior to the course notions on the concept of chemical solutions, how to express concentration of solutions, chemical formulas and mathematical operations. The statement did not include data on the volumes of the mixed solutions, and was written looking for a “probable answer”.

Question 2: this question uses verbal-procedural language. It is stated in everyday language: the procedure was described in detail and mass and volume values were explicit, instead of providing concentration units. In this way, we intended to assess whether the mental representation of the preparation of a real solution could help students choose the correct option. It was expected that this format could be meaningful even for students with little knowledge of chemistry. The answer options were similar to those of the previous question, and also included the “none of the above answers is right” as a distractor.

Question 3: this question uses visual language. It contains colored images, and does not require numerical calculations. This format aims to present a visual analogy between average color and average concentration. This analogy alludes to a representational model in which the mixture of two parts produces an outcome with an intermediate color, not an additive one.

It was expected that this question was answered using intuition and visual records rather than chemical knowledge. While this analogy is not always accurate, we are assuming that many everyday experiences such as preparing drinks from concentrate juice or mixtures of watercolors to achieve desired hues could be significant for students.


Table 1 presents the distribution of answers of the 532 students who participated in the survey. Question 1 that used the chemical formulas language had the lowest percentage of students who answered it correctly (44%), however, this was the question answered by the highest percentage of students (only 4.9% did not answer it). Question 2, which uses verbal-procedural language, was the second with the lowest percentage of correct answers (45.9%). In other words, questions 1 and 2 that did not use visual language are the ones that obtained the least correct answers. In contrast, it was noted that the question 3 was answered correctly by a higher percentage of students (65.4%). This question uses visual language and presents an analogy between mixture of hues and mixture of concentrations. It should be noted that only 162 students (30.5%) answered all three questions correctly. Chi-Square Test determined no significant difference between the values obtained for the answers of question 1 and 2 [χ 2 (N = 532) = 0.20,920, p < 0.01]. However, the answers to questions 1 and 3 do show significant differences [χ 2 (N = 532) = 223,298, p < 0.01], the same as the answers to questions 2 and 3 [χ 2 (N = 532) = 182,702, p < 0.01].

Table 1:

Distribution of answers (n = 532).

Answers of students Question 1 Question 2 Question 3
Incorrect answers 51.1% 47.2% 25.4%
Correct answers 44% 45.9% 65.4%
Student does not answer 4.9% 7% 9.2%

Examples of incorrect answers from students

Question 1: (using formula language and expressions of chemical concentration):

The most frequent error was found in the answers in which the “sum of concentrations” (option (b)) was chosen.

A typical example of written justification is: ‘if solutions consist of the same solute mixed with the same solvent, concentrations must be added.’

Question 2: (using verbal-procedural language):

Some students reasoned correctly, but possibly made mistakes by not knowing the proper expression in the language of chemical concentration in g/L. This situation is typically noted in the answer justifications (c) of this question.

Fifteen of the respondents said that ‘when the contents of the beaker A and B are dumped in the same container, the final concentration would be 600 g of sugar into 2 L’. In turn, in 18 cases, it was clarified that ‘none is correct since to calculate the concentration, I have to divide the solute mass by the volume, but as the volume is not given, I cannot do it.’

Question 3: (using visual language):

In the wrong answers (b), appear justifications in which some “mathematical” arguments are used without any conceptual support; two examples are reproduced below:

  1. ‘I calculated the relative mass, which is 158, and I reasoned that if there are 158 g of KMnO4 in 1000 g, how much will there be in 1 g of KMnO4. This resulted in 6.33 g, and then I repeated the same operation for the second solution with 6 g, which gave me 37.8 g. Hence, I took equal parts of each solution (i.e., half of the grams I calculated). 3.165 g of the first one, and with the same procedure, 18.9 g of the second one. This gave me 22.06 g for the dilute solution, so it seems to me that the container is the b’.

  2. Here the student starts by calculating the molar mass, a very usual procedure in this kind of problems but unnecessary in this particular case. He then tries to use a numerical procedure in order to obtain a ratio, introducing for this purpose a data value not present in the problem statement (1000 g). Since his procedure does not have any chemical basis, his conclusion is invalid.

  3. ‘For example, 1 g of each solution will be mixed. The answer is the solution B, because if the same amount of solvent is used for both (water), that is, twice the initial amount; and the amount of solute is twice that in the solution B, the final solution will have the same initial color than B’.

  4. In this case, although the student does consider explicitly the solute and the solvent, she does not take into account the fact that the concentrations in each container differ. In other words, she does not realize that in 1 g of each solution there are different solvent volumes (and masses). Then, she incorrectly reasons as if she could simply double one of the mixture components.

  5. Among the errors in answering this question, many answers that said the options (d) and (e) were observed. Justifications found show the existence of mental models about the darker color predominates over clear color. Examples of such justifications are:

  6. Option (d): ‘dark colors predominate over light colors, in the combinations of the same amount.’

  7. Option (d) ‘because it is the dominant.’

  8. Option (e), ‘the color is not lost.’

  9. Option (e): ‘The resulting solution will have the color of the choice “d”, as equal parts of each solution were mixed, and as the solution B does not affect the D, it remains with the same color. For solution B changes its color, its concentration should be greater than the solution D.’

Interestingly, 88, 74 and 21% of the incorrect answers to questions 1, 2 and 3 respectively reflect the adding up the concentrations of the solutions.


Taking into account the justifications written by students to make their answers, categories of analysis could be generated, regarding their heuristic (Talanquer, 2006) —forms of reasoning—. Such misconceptions were classified into different categories:

  1. Those produced by the conviction that the use of mathematics to solve any situation is necessary: they appear in students that appeal to mathematical operations of any kind, often without any chemical sense (e.g., Question 3, examples i, ii).

  2. Those from alternative models to the expected ones (e.g., Questions 3, examples iii, iv, v, vi): certain operations of daily life, such as using dark colors on walls or hair dyes can generate models in freshmen not consistent with those expected for problem solving of solutions concentration.

In the wrong arguments, there are always certain words or phrases that refer to correct concepts taught, but were memorized or not significantly learnt. This partial knowledge may hinder correct reasoning and lead to wrong conclusions. For example, students correctly calculate relative masses, reason about solutes and solvents, but they do not take it into account or confuse masses and volumes of the components of the solutions.

As we previously described in Methodology, question 1 presents a typical chemistry class problem, while question 2 is of a verbal-procedural nature, since it raises an approach typical of the chemistry laboratory. Instead, question 3 appeals to a visual language, which can be related to everyday knowledge. However, the languages of the questions on the same topic represent by themselves different complexities for resolution (Gardner, 1983).

The use of analytical languages —with mathematical and chemical formulas—, despite arriving at conceptually wrong answers, can be seen in Table 1, where the questions that only require analytical or verbal-procedural calculations presented a lower number of students who did not solve them. This observation was also made in a previous investigation (Bekerman et al., 2016), in which simulations were used to solve solution concentration problems. In this case, a group of students solved the problem numerically, without taking into account the physical quantities involved.

The lessons received by the students who participated in this research are based on the analytical resolution of chemistry solution problems and mixtures of chemistry solutions. In other words, they are fundamentally trained in the use of the level of symbolic representation (Johnstone, 1991), by exclusively using formula language and mathematical expressions of chemical concentration. Therefore, students are much more familiar with the use of mathematical and formula language, since they do not have laboratory classes. Taking the chemical conceptualisation into account would lead directly to the answer, even though these questions include mathematical and chemical formula language. Nevertheless, the presence of this language led them to move away from the most obvious and correct resolution, and to consider that this was a problem to be solved analytically.

On the other hand, question 3, which uses visual language, was answered correctly by a greater number of students, showing a statistically significant difference with the other two questions. This significant difference between the number of correct answers to the mathematical or formula and procedural language questions and to the questions with visual language would allow inferring that the use of this language brings students closer to a better understanding of the situation. In other words, the macroscopic level of representation facilitates in this case the correct resolution for those students who manage to detach themselves from the mathematical and symbolic language they received in their classes. However, a future instance of interviews would be necessary to have greater certainty of the influence of this language as a facilitator of the student’s formal reasoning.

These results are in line with other authors who also affirm that there is a trend to apply mathematical calculations when chemical problems are addressed, although such calculations are not necessary (Nyachwaya, Warfa, Roehrig, & Schneider, 2014). Thus, obtaining a numerical result would be considered appropriate by the students, with no analysis of the conceptual significance of the number derived, or of the qualitative-conceptual relationships between the variables involved (Raviolo, Farré, & Traiman Schroh, 2021). Zoller (2002) found that the students had a recipe they used to solve problems. They applied memorized algorithms which they believed would always get them the ‘right answer’. Zoller concluded that ‘students have been conditioned to think algorithmically’. According to Raviolo et al. (2021), this type of resolution, as well as the numerical ones, can hide the lack of conceptual understanding; for example, students who use a numerical resolution did not conceptually distinguish between number of moles (extensive property) and Molarity (intensive property) (Raviolo et al., 2021). In our work, we have found similar results. Many students answered that when mixing two solutions of different concentration another with a higher concentration was obtained. Hence, they would also be confusing an extensive property expressed, in this case, in grams, with an intensive one, expressed in grams per liter. The most frequent erroneous justifications found in the answers was to consider the sum of the concentrations of the solutions, as if it were an extensive property (see Results). This is another evidence that students adhere to formula language without understanding the meaning of the result, for which procedural or visual language would help to understand clearly.

Similarly, Dahsah and Coll (2007, 2008 observed that, when solving, for example, stoichiometry exercises, students use algorithms, although they do not fully understand the concepts involved. Likewise, Bodner (1987) calls for a differentiation between solving exercises using an algorithm and problem solving, which is a more complex process, and Izquierdo-Aymerich (2005), affirms that for the resolution of chemistry problems, emphasis is placed on the use of chemical formulas and equations that are memorized.

The resolution of exercises and problems demands the integration of the different levels of representation, which are intimately connected (Lemke, 1998; Sirhan, 2007) and that correspond to different languages of chemistry. Consequently, we agreed with Wu et al. (2001), cited in Jansoon, Coll, and Somsook (2009), the need to promote learning that combines laboratory practices and conceptual understanding, and consequently evaluate, beyond the obtaining ‘a correct numerical answer […] that fails to genuinely probe the students’ understanding’ (Jansoon et al., 2009). In this same sense, Costu (2010) and Fajardo and Bacarrisas (2017) suggest that teachers should focus on conceptual understanding by students.

Conclusions and final remarks

Taking into account all the above considerations, the following didactic question arises: how to show students that the mixture of solutions generates a solution of intermediate concentration to source solutions? This means: a) how can the teaching of this subject in the class be addressed, not just from an intuitive explanation, but rooted in the theory, b) how can we achieve that the resolution of these problems —and those of other topics— lead students to appeal to the different thinking models typical of chemistry, so that the mathematical resolution is accompanied by other practical and conceptual approaches. Similar to authors cited above, we strongly believe that adequate guidance that includes an empirical and analytical approach could be very useful, especially for students who tend to perform calculations to solve chemical situations.

Corresponding author: Manuel Alonso, Universidad de Buenos Aires, Ciclo Básico Común, Departamento de Ciencias Biológicas, Ramos Mejía 841 C1405CAE, Buenos Aires, Argentina, E-mail:

Award Identifier / Grant number: UBACyT 20020170100469BA

  1. Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: This work was supported by grant UBACyT 20020170100469BA of the University of Buenos Aires.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.


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Received: 2021-08-31
Accepted: 2022-06-16
Published Online: 2022-08-04

© 2022 the author(s), published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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