Question

Consider 100.0 -g samples of two different compounds consisting only of carbon and oxygen. One compound contains \(27.2 \mathrm{g}\) of carbon and the other has \(42.9 \mathrm{g}\) of carbon. How can these data support the law of multiple proportions if 42.9 is not a multiple of \(27.2 ?\) Show that these data support the law of multiple proportions.

Short answer

The data supports the Law of Multiple Proportions since the simplified mass ratios of carbon in the two compounds are in a whole number ratio (1:3). Even though the ratio of 42.9 to 27.2 is not a whole number, the simplified mass ratios (1:1 for the second compound and 1:3 for the first compound) follow the law.

Step-by-step solution

Determine the mass of oxygen in each compound

We are given 100.0 g samples of two different compounds consisting only of carbon and oxygen. The first compound contains 27.2 g of carbon, and the second compound contains 42.9 g of carbon. To find the mass of oxygen in each compound, we subtract the mass of Carbon from the total mass. For the first compound: Mass of Oxygen = Total mass - Mass of Carbon = 100.0 g - 27.2 g = 72.8 g For the second compound: Mass of Oxygen = Total mass - Mass of Carbon = 100.0 g - 42.9 g = 57.1 g

Calculate the mass ratios of carbon and oxygen for each compound

Now we need to determine the ratio of the mass of Carbon to the mass of Oxygen in each compound. For the first compound: Carbon:Oxygen = 27.2 g : 72.8 g For the second compound: Carbon:Oxygen = 42.9 g : 57.1 g

Simplify the ratios to the simplest whole number ratios

Now, we must simplify the Carbon:Oxygen ratios to the simplest whole number ratios. For the first compound (27.2 g Carbon : 72.8 g Oxygen), we divide both numbers by 27.2: Carbon:Oxygen = 1 : (72.8 ÷ 27.2) ≈ 1 : 2.68 ≈ 1 : 3 (rounded to the nearest whole number) For the second compound (42.9 g Carbon : 57.1 g Oxygen), we divide both numbers by 42.9: Carbon:Oxygen = 1 : (57.1 ÷ 42.9) ≈ 1 : 1.33 ≈ 1 : 1 (rounded to the nearest whole number)

Verify the law of multiple proportions

Having simplified the Carbon:Oxygen ratios to the simplest whole number ratios for both compounds, we can now verify if these data support the law of multiple proportions. We will compare the Carbon mass ratios for both compounds and see if they are in whole number ratios. The mass of Carbon in the first compound combines with Oxygen in a 1:3 ratio, and the mass of Carbon in the second compound combines with Oxygen in a 1:1 ratio. The ratio of the Carbon masses in the two compounds is: \( \frac{42.9}{27.2} ≈ 1.58 \) While the ratio of 42.9 to 27.2 is not a whole number, we can look at the ratios between the simplified mass ratios of Carbon: \( \frac{1:1}{1:3} = \frac{1}{3} \) The ratio between the simplified mass ratios of Carbon is 1:3, which is a whole number ratio. Therefore, these data support the Law of Multiple Proportions.

Key concepts

Mass Ratios in Chemistry

Understanding mass ratios is essential in the field of chemistry, particularly when examining the composition of chemical compounds. A mass ratio compares the amount of one substance to the amount of another within a compound. For instance, in the exercise concerning the two carbon-oxygen compounds, the focus is on the mass ratios of carbon to oxygen. The law of multiple proportions, devised by John Dalton, states that when two elements combine to form more than one compound, the masses of one element that combine with a fixed mass of the other are in ratios of small whole numbers.
To grasp this concept, using the example provided, the mass of oxygen is calculated by subtracting the mass of carbon from the total mass of the compound. We then compare these masses to determine the ratio. Despite initial appearances, the law of multiple proportions is confirmed when these ratios simplify to whole numbers and show a straightforward relationship between the two compounds—one that can be described by small whole numbers, affirming Dalton's principle.

Chemical Compound Composition

Elemental Significance

Every chemical compound has a specific and fixed composition by mass. This is related to the law of definite proportions, which states that a chemical compound always contains exactly the same proportion of elements by mass. This ties directly into the exercise, where we examined compounds of carbon and oxygen, each with a specific mass of carbon. The remaining mass was attributed to oxygen, adhering to the proportions dictated by the unique identity of the compound.

Verification and the Law

An extension of understanding this concept is verifying the law of multiple proportions. By doing so, students enhance their comprehension of chemical formulas and the underlying stoichiometry involved. It is fascinating how the law holds true even when the mass ratios of the elements are not apparent whole numbers at first glance but reveal themselves upon deeper analysis, as seen in the calculation steps provided.

Stoichiometry

Stoichiometry is the branch of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. It is also concerned with the ratios in which elements combine to form compounds. This touches on the heart of our exercise, where the ratio of carbon to oxygen in two different compounds was scrutinized. The stepwise approach calculates these ratios and then simplifies them into whole numbers—a process emanating from stoichiometrical principles.
Moreover, understanding stoichiometry is vital for students when predicting the quantities of substances consumed and produced in a given reaction. It cultivates a methodical way of thinking about chemical reactions, not just to support the law of multiple proportions but also as a gateway to exploring more complex chemical equations and reactions.