Question

Consider the generic chemical equation aA \(+\mathrm{bB} \rightarrow\) \(\mathrm{cC}+\mathrm{dD}\) (where a, b, \(\mathrm{c},\) and \(\mathrm{d}\) represent coefficients for the chemicals \(A, B, C,\) and \(D,\) respectively). a. How many possible values are there for "c"? Explain your answer. b. How many possible values are there for "c/d"? Explain your answer.

Short answer

a. There are infinitely many possible values for "c" since it can be any positive integer. b. There are infinitely many possible values for "c/d" as well, as both "c" and "d" can take any positive integer value and their ratio can be represented uniquely as a simplified fraction \(\frac{p}{q}\), where p and q are coprime integers.

Step-by-step solution

Understand the given chemical equation and its coefficients

The coefficients in the chemical equation represent the number of molecules or atoms of each reactant and product that are involved in the chemical reaction. In a balanced chemical equation, the number of atoms of each element must be equal on both sides of the equation. Coefficients must be positive integers as they represent the quantity of molecules or atoms in the reaction.

Determine possible values for "c"

Since "c" represents the coefficient of chemical C in the balanced equation, it must be a positive integer. There are infinitely many positive integers. Therefore, there are infinitely many possible values for "c".

Determine possible values for "c/d"

In this part, we are looking for the possible values for the ratio of "c" to "d". Since both "c" and "d" can take any positive integer value, their ratio can have infinitely many possibilities. However, by simplifying or reducing the ratios, we can represent each ratio uniquely as a fraction in the form of \(\frac{p}{q}\), where p and q are integers without any common factors other than 1 (i.e., they are coprime). For example, let's consider some possible ratios: 1. If \(c = 2\) and \(d = 4\), then \(\frac{c}{d} = \frac{2}{4}\) which simplifies to \(\frac{1}{2}\). 2. If \(c = 4\) and \(d = 6\), then \(\frac{c}{d} = \frac{4}{6}\) which simplifies to \(\frac{2}{3}\). In each case, the simplified ratio represents a unique relationship between the coefficients "c" and "d". Since we have infinitely many possible pairs of "c" and "d", there are also infinitely many possible values of their ratio "\(\frac{c}{d}\)". So, the answer to part a is that there are infinitely many possible values for "c," and the answer to part b is that there are infinitely many possible values for "c/d".

Key concepts

Balanced Chemical Equation

The cornerstone of any chemical reaction analysis is the balanced chemical equation. But why is it so vital to balance these equations? Quite simply, it's all about the law of conservation of mass. This law states that mass cannot be created or destroyed in a chemical reaction. In terms of a chemical equation, it means that the number of atoms for each element must be equal on both sides of the equation.

Take water formation, for instance: when hydrogen gas reacts with oxygen gas, the equation is written as \(2H_2 + O_2 \rightarrow 2H_2O\). This equation shows that two molecules of hydrogen and one molecule of oxygen produce two molecules of water. We can't have half a molecule of oxygen react, hence the coefficients must be whole numbers. They also ensure that the number of hydrogen atoms equals on both sides (4 hydrogen atoms) as does the number of oxygen atoms (2 oxygen atoms).

When discussing possible values for a coefficient in a generic chemical equation like aA + bB → cC + dD, the coefficient must be chosen so that the equation is balanced. Thus, 'c' can represent any positive integer, but only specific values will balance the equation with the given coefficients for A, B, and D.

Stoichiometry

Stoichiometry is essentially the recipe for a chemical reaction, detailing the exact amounts of reactants needed to produce a certain amount of product. It's what bakers do with their recipes, but instead of cups and teaspoons, chemists use moles. This is crucial, as calculating the amounts of substances consumed and produced in reactions allows for the prediction of yields and the planning of chemical syntheses.

Regarding stoichiometry in a balanced equation like aA + bB → cC + dD, it determines the mole ratios needed for the reactants to produce the desired quantity of products without wasting any material. For example, if an equation shows that one mole of A reacts with two moles of B to produce three moles of C (1A + 2B → 3C), we can predict how much B we’ll need if we start with a specific amount of A, and vice versa.

The coefficient 'c' can vary, but it must relate to the other coefficients in a manner that obeys the stoichiometric relationships defined by the balanced chemical equation. The correct stoichiometry is paramount because otherwise, we risk reactions not going to completion or having excess reactants.

Mole Ratio

When it comes to mole ratios, think of them as the deep connections between reactants and products in a chemical equation. They tell us the 'if-then' story of the reaction: if you have a certain amount of one substance, then you can expect a certain amount of another. These ratios can be derived from the coefficients of a balanced chemical equation. In the example of water formation (2H_2 + O_2 → 2H_2O), the mole ratio between H_2 and O_2 is 2:1, which means for every two moles of hydrogen, one mole of oxygen is used.

The ratio of the coefficients, like 'c/d', is essential in showing how much of one product is created relative to another. Although there are infinitely many possibilities for 'c' and 'd' if we consider them individually on a theoretical level, in practice, they are not arbitrary; they must reflect the balanced equation that governs the reaction. For example, if 'c' was 4 and 'd' was 2, the mole ratio of C to D would be 2:1. Yet, every pair of values for 'c' and 'd' has its unique ratio when simplified to the lowest terms, ensuring that mole ratios are exact and meaningful in the context of the reaction.