Question

\- Molecules of oxygen gas have been very difficult to find in space. Recently the European Space Agency's Herschel space observatory has found molecular oxygen in the Orion star-forming complex. It is present at very low concentrations, 1 molecule of oxygen per every million \(\left(1.0 \times 10^{6}\right)\) molecules of hydrogen, but it does exist. How many molecules of oxygen would exist in a \(3.0\)-g sample of hydrogen gas?

Short answer

There are \(9.033 \times 10^{17}\) oxygen molecules in a 3.0 g sample of hydrogen gas.

Step-by-step solution

Determine the number of moles of hydrogen

First, find the number of moles of hydrogen in a 3.0 g sample. The molecular weight of hydrogen \(\text{H}_2\) is 2.0 g/mol. Use the formula: \[ \text{moles of hydrogen} = \frac{\text{mass of hydrogen}}{\text{molecular weight of hydrogen}} = \frac{3.0 \ \text{g}}{2.0 \ \text{g/mol}} = 1.5 \ \text{mol} \]

Calculate the number of hydrogen molecules

Use Avogadro’s number \(6.022 \times 10^{23} \ \text{molecules/mol}\) to find the number of hydrogen molecules in 1.5 moles: \[ \text{number of hydrogen molecules} = 1.5 \ \text{mol} \times 6.022 \times 10^{23} \ \text{molecules/mol} = 9.033 \times 10^{23} \ \text{molecules} \]

Relate hydrogen molecules to oxygen molecules

Given that there is 1 molecule of oxygen for every million \(1.0 \times 10^{6}\) molecules of hydrogen, the number of oxygen molecules is: \[ \text{number of oxygen molecules} = \frac{9.033 \times 10^{23} \ \text{hydrogen molecules}}{1.0 \times 10^{6}} = 9.033 \times 10^{17} \ \text{oxygen molecules} \]

Key concepts

moles and molecules calculation

To solve the problem, we first need to understand moles and how they relate to the number of molecules.
A mole is a unit used to count very large quantities of tiny entities like atoms or molecules. It's like a 'dozen' but instead of 12, a mole represents \(6.022 \times 10^{23}\) entities.
When we calculate moles, we use the molecular weight of the substance. In this case, hydrogen (\text{H}_2\text{) has a molecular weight of 2.0 g/mol.
Given 3.0 g of hydrogen, we use the formula:

\[ \text{moles of hydrogen} = \frac{\text{mass of hydrogen}}{\text{molecular weight of hydrogen}} = \frac{3.0 \text{g}}{2.0 \text{g/mol}} = 1.5 \text{mol} \]

So, in our 3.0 g sample, we have 1.5 moles of hydrogen.
Now, we need to convert these moles into actual molecules, which is where Avogadro’s number comes into play.

Avogadro's number

Avogadro's number is a key concept in chemistry. It defines the number of atoms, ions, or molecules in one mole of a substance. This number is \(6.022 \times 10^{23}\).
To find the number of hydrogen molecules in our sample, we multiply the number of moles by Avogadro’s number:

\[ \text{number of hydrogen molecules} = 1.5 \text{mol} \times 6.022 \times 10^{23} \text{molecules/mol} = 9.033 \times 10^{23} \text{molecules} \]

We now know there are \(9.033 \times 10^{23}\) hydrogen molecules in our 3.0 g sample.
Next, we connect this to the number of oxygen molecules.

molecular proportions in space

The exercise states a proportion: 1 oxygen molecule exists for every million ( \(1.0 \times 10^{6}\) ) hydrogen molecules. This proportion will help us calculate the number of oxygen molecules.
Using the hydrogen molecule count from before, we apply the given proportion:

\[ \text{number of oxygen molecules} = \frac{9.033 \times 10^{23} \text{hydrogen molecules}}{1.0 \times 10^{6}} = 9.033 \times 10^{17} \text{oxygen molecules} \]

Hence, in a 3.0 g sample of hydrogen, there are \(9.033 \times 10^{17}\) oxygen molecules.
Understanding this application of ratios and molecular counting is vital for solving similar problems in both academic and real-world contexts.