Question

Samples of pure carbon weighing \(3.62,5.91,\) and \(7.07 \mathrm{g}\) were burned in an excess of air. The masses of carbon dioxide obtained (the sole product in each case) were \(13.26,21.66,\) and \(25.91 \mathrm{g},\) respectively. (a) Do these data establish that carbon dioxide has a fixed composition? (b) What is the composition of carbon dioxide, expressed in \% C and \% O, by mass?

Short answer

Firstly, after calculating ratios of mass of CO2 to mass of C for each trial, they come close to 3.66, attesting the validity of the law of definite proportions and hence the fixed composition of carbon dioxide. Secondly, the mass percentage composition of carbon dioxide comes out be approximately 27.29% Carbon(C) and 72.71% Oxygen (O).

Step-by-step solution

Confirming the law of definite proportions

According to the law of definite proportions, if carbon dioxide has a fixed composition, the ratio of the mass of carbon dioxide produced to the mass of carbon used should be the same in all three trials. To confirm this, calculate the ratio for each trial: \[ \text{Ratio} = \frac{\text{Mass of } CO_2}{\text{Mass of Carbon}} \] For the first trial, this will be \(\frac{13.26}{3.62}\), the second trial will be \(\frac{21.66}{5.91}\), and the third trial will be \(\frac{25.91}{7.07}\). If these values are the same or very close to one another, then carbon dioxide does have a fixed composition.

Calculating the composition of carbon dioxide in % C and % O by mass

To confirm the composition of carbon dioxide by mass, understanding that a molecule of carbon dioxide (CO2) consists of one atom of carbon and two atoms of oxygen is necessary. Total molecular mass can be calculated by adding the atomic masses of its constituents (from the periodic table): carbon has a mass of 12.01 g/mol and oxygen has a mass of 16.00 g/mol. So total mass of carbon dioxide (CO2) comes to \(12.01+2*(16.00) = 44.01 \mathrm{g/mol}\). \nThen we calculate the mass percent of Carbon (C) in Carbon dioxide (CO2): \[ \text{% C} = \left( \frac{\text{Carbon's mass}}{\text{Total mass}} \right) * 100\% = \left(\frac{12.01}{44.01}\right) * 100\%\] Similarly, calculate the mass percent of Oxygen (O) in Carbon dioxide (CO2): \[ \text{% O} = \left( \frac{\text{Oxygen's mass}}{\text{Total mass}} \right) * 100\% = \left(\frac{2*16.00}{44.01}\right) * 100\%\]

Key concepts

Fixed Composition

The concept of fixed composition refers to the law of definite proportions. This law states that a chemical compound always contains exactly the same proportion of elements by mass. Let's use carbon dioxide as an example. No matter the source or the amount of carbon dioxide you have, the ratio of the mass of carbon to the mass of oxygen remains consistent.
In the exercise, pure carbon was burned, resulting in different masses of carbon dioxide. Despite using different carbon masses (3.62 g, 5.91 g, and 7.07 g), each results in varying amounts of carbon dioxide (13.26 g, 21.66 g, and 25.91 g respectively).
This data can be analyzed using the mass ratio:

• For the first trial: \(13.26 / 3.62\)
• For the second trial: \(21.66 / 5.91\)
• For the third trial: \(25.91 / 7.07\)

By looking at these ratios, if they are nearly the same, it shows carbon dioxide's composition remains fixed. This confirms the law of definite proportions in carbon dioxide.

Carbon Dioxide

Carbon dioxide is a chemical compound composed of one carbon atom and two oxygen atoms. It is represented by the chemical formula \(CO_2\). Carbon dioxide is a significant component in the atmosphere and plays a vital role in processes such as photosynthesis and the carbon cycle.
The fixed composition of carbon dioxide implies that, regardless of where it is found in nature or produced artificially, its structure remains identical. This means one carbon atom is always bonded to two oxygen atoms.
Moreover, the formation process of carbon dioxide is straightforward. When carbon burns in oxygen-rich environments, it reacts with oxygen to form carbon dioxide. The equation for this reaction is:

• \(C + O_2 \rightarrow CO_2\)

This simplistic reaction underscores why carbon dioxide behaves under the law of definite proportions, maintaining its fixed composition.

Mass Percent Composition

Mass percent composition is a way of expressing the concentration of an element in a compound. It tells us the relative masses of each element in the compound compared to its total mass.
For carbon dioxide, the molecular mass is derived from the atomic masses. Carbon has an atomic mass of 12.01 g/mol, while each oxygen atom is 16.00 g/mol. Therefore, the molecular weight of carbon dioxide is \(12.01 + 2 \times 16.00 = 44.01 \text{ g/mol}\).

Now, let's determine the mass percent of each element:

• For Carbon, the mass percent is:\[\text{% C} = \left( \frac{12.01}{44.01} \right) \times 100\%\]
• For Oxygen, the mass percent is:\[\text{% O} = \left( \frac{2 \times 16.00}{44.01} \right) \times 100\%\]

These percentages show how much of a compound's mass is one element, reaffirming that carbon dioxide maintains its consistent chemical makeup by mass wherever it's found.