Question

How many molecules of ethane \(\left(\mathrm{C}_{2} \mathrm{H}_{6}\right)\) are present in \(0.334 \mathrm{~g}\) of \(\mathrm{C}_{2} \mathrm{H}_{6} ?\)

Short answer

There are approximately \(6.69 \times 10^{21}\) molecules of ethane in \(0.334\, \text{g}\) of \(C_2H_6\).

Step-by-step solution

Calculate the Molar Mass

First, calculate the molar mass of \ \( C_2H_6 \ \). Ethane consists of 2 carbon atoms and 6 hydrogen atoms. The molar mass of carbon \ (C) \ is approximately 12.01 g/mol, and that of hydrogen \ (H) \ is approximately 1.01 g/mol. Thus, the molar mass of ethane is: \[ \text{Molar mass of } C_2H_6 = 2(12.01) + 6(1.01) = 30.08 \, \text{g/mol} \]

Find the Number of Moles

Use the formula for moles, which is \(( \text{moles} = \frac{\text{mass in grams}}{\text{molar mass}})\). Plug in the known values:\[ \text{Moles of } C_2H_6 = \frac{0.334 \, \text{g}}{30.08 \, \text{g/mol}} = 0.0111 \, \text{moles} \]

Calculate the Number of Molecules

To find the number of molecules, use Avogadro's number, which is \( 6.022 \times 10^{23} \) molecules/mol. Multiply the number of moles by Avogadro's number:\[ \text{Number of molecules} = 0.0111 \, \text{moles} \times 6.022 \times 10^{23} \, \text{molecules/mol} \approx 6.69 \times 10^{21} \, \text{molecules} \]

Key concepts

Avogadro's Number

Avogadro's Number is a fundamental constant that plays a crucial role in chemistry. It is the key to linking the macroscopic scale of grams and liters to the microscopic scale of atoms and molecules. This number, named after the scientist Amedeo Avogadro, is defined as the number of atoms, ions, or molecules in one mole of a substance. The value is approximately \( 6.022 \times 10^{23} \). This means that one mole of any substance contains exactly this number of constituent particles.

Understanding Avogadro's Number helps in converting between the mass of a substance and the number of molecules or atoms it contains. When you calculate how many molecules are in a given mass of a substance, like ethane, you're essentially scaling up from the individual molecular level to something we can measure and observe in the real world. For our exercise, once you know the number of moles of ethane, multiplying it by Avogadro's Number gives you the total number of ethane molecules present.

Number of Moles

The concept of moles is central to the world of chemistry, providing a bridge between atomic scale measurements and everyday quantities. A mole is defined as the amount of a substance that contains the same number of entities (atoms, molecules, ions, etc.) as there are in 12 grams of pure carbon-12. This number of entities is exactly Avogadro's Number, which is \( 6.022 \times 10^{23} \).

The number of moles allows chemists to work with the massive quantities of entities involved in real-world chemical reactions. To find the number of moles in a substance, divide the mass of the sample by its molar mass. For example, in our ethane exercise, you take the given mass of ethane, 0.334 grams, and divide it by the calculated molar mass of ethane (30.08 g/mol). This calculation tells you how many moles, or atoms-worth clusters, of ethane you have.

Ethane

Ethane is a simple hydrocarbon with the chemical formula \( C_2H_6 \). It is part of the alkane series, which contains only hydrogen and carbon atoms connected by single bonds. Ethane is a colorless, odorless gas at room temperature, commonly found in natural gas and used widely in the chemical industry.

The molar mass is an important factor when working with ethane, especially in chemical calculations like the one in the exercise. The molar mass of ethane is calculated based on its atomic composition: 2 carbon atoms (each contributing around 12.01 g/mol) and 6 hydrogen atoms (each contributing about 1.01 g/mol). This totals to 30.08 g/mol. Knowing this allows us to convert between grams of ethane and moles, enabling us to use Avogadro's Number to find the number of molecules from a given mass of ethane.

In practical applications, understanding these conversions is vital for everything from producing chemicals efficiently to ensuring the correct dosage of pharmaceutical compounds. Ethane's simplicity as a molecule makes it a great example for learning fundamental chemistry concepts like moles and molecular calculations.