Problem 46 - General Chemistry

Question

Nitrous oxide, \(\mathrm{N}_{2} \mathrm{O}\), has been used as a dental anesthetic. The average speed of an \(\mathrm{N}_{2} \mathrm{O}\) molecule at \(25^{\circ} \mathrm{C}\) is \(379 \mathrm{~m} / \mathrm{s}\). Calculate the kinetic energy (in joules) of an \(\mathrm{N}_{2} \mathrm{O}\) molecule traveling at this speed.

Step-by-Step Solution

Answer

The kinetic energy is approximately \( 5.269 \times 10^{-21} \mathrm{~J} \).

Step 1: Identify Given Values

We are given the following values:- Average speed of \( \mathrm{N}_2\mathrm{O} \) molecule, \( v = 379 \mathrm{~m/s} \).- The molar mass of \( \mathrm{N}_2\mathrm{O} \) is approximately \( 44.01 \mathrm{~g/mol} \). This needs to be converted to kg to use in our calculations. Recall that \( 1 \mathrm{~g} = 0.001 \mathrm{~kg} \), so the molar mass is \( 44.01 \times 10^{-3} \mathrm{~kg/mol} \).

Step 2: Calculate Molecular Mass in kg

Find the mass of a single \( \mathrm{N}_2\mathrm{O} \) molecule. First, find the mass of one mole of \( \mathrm{N}_2\mathrm{O} \), which is \( 44.01 \times 10^{-3} \mathrm{~kg/mol} \). The mass of a single molecule is \( m = \frac{44.01 \times 10^{-3}}{6.022 \times 10^{23}} \mathrm{~kg} \), where \( 6.022 \times 10^{23} \) is Avogadro's number.

Step 3: Use Kinetic Energy Formula

The kinetic energy of a molecule is given by the formula \( KE = \frac{1}{2}mv^2 \). We already have \( v = 379 \mathrm{~m/s} \) and the expression for \( m \) from the previous step. Substitute these values into the formula to find the kinetic energy.

Step 4: Plug in the Values

Substitute the values: - \( m = \frac{44.01 \times 10^{-3}}{6.022 \times 10^{23}} \mathrm{~kg} \).- \( v = 379 \mathrm{~m/s} \).Calculate \( KE = \frac{1}{2} \left( \frac{44.01 \times 10^{-3}}{6.022 \times 10^{23}} \right) (379)^2 \).

Step 5: Calculate the Result

Perform the calculation:- First, calculate the mass, \( m = 7.308 \times 10^{-26} \mathrm{~kg} \).- Then, substitute \( m \) and \( v \) into the formula: \[ KE = \frac{1}{2} \times 7.308 \times 10^{-26} \times (379)^2 \].- Finally, find that \( KE \approx 5.269 \times 10^{-21} \mathrm{~J} \).

Key Concepts

Understanding Molecular Speed

Molecular speed refers to how fast individual molecules are moving. Thank goodness, we're not entirely blind when it comes to measuring such speeds! The average molecular speed is essential for calculating the kinetic energy of a molecule.
For our nitrous oxide ( 2O) molecule at 25°C, the average speed is given as 379 m/s. It's crucial to note that this is just the average; molecules in a sample will have a range of speeds.
When we talk about molecular speed, we envision lots of tiny, tiny molecules darting around in different directions, each with an average speed that scientists can calculate and use in further equations.

Calculating Molar Mass

Molar mass is a term that helps us bridge the gap between the micro-world of atoms and molecules, and the macro-world where we weigh things out in grams or kilograms. Specifically, it's the mass of one mole of a substance, measured in grams per mole (g/mol).
For nitrous oxide ( 2O), the molar mass is found by adding up the atomic masses of all the atoms within a molecule of the substance. In this case, it is about 44.01 g/mol. But for our calculations involving kinetic energy, we need to convert this into kilograms per mole. Keep in mind, the conversion is straightforward: multiply by 0.001 to convert grams to kilograms. This means our molar mass becomes 44.01 × 10^{-3} kg/mol.
Why kilograms? Well, in physics, using the same units is critical, and kinetic energy calculations are typically done in kilograms.

The Role of Avogadro's Number

Avogadro's number is a very important concept when dealing with molecules and the calculations involving them. It is a constant, roughly equal to 6.022 × 10^{23}. What does this number mean? It represents the number of atoms or molecules in one mole of a substance.
In essence, when we are trying to determine the mass of a single molecule of nitrous oxide, we rely heavily on Avogadro's number. It allows us to divide the molar mass by the number of molecules in a mole to find the mass of one molecule.
This conversion is essential because it allows us to think in terms of individual molecules, making it possible to calculate the kinetic energy at this precise level. So the number truly helps us scale down from the macroscopic world of moles to the tiny, real-world granularity of single molecules.

Delving into Molecular Calculations

Molecular calculations might sound like something you'd need a supercomputer for, but fear not! These calculations can be broken down and simplified.
By grasping the average speed, the molar mass, and Avogadro’s number, you can compute other pertinent quantities, like the kinetic energy of a single molecule.
The formula for kinetic energy is: \[ KE = \frac{1}{2} mv^2 \] This shows that kinetic energy is directly tied to both the mass and speed of the molecule. In our exercise, by substituting the mass (derived using Avogadro's number) and molecular speed, you can find the precise kinetic energy in joules of a single 2O molecule. Following through these calculations successfully links together our understanding of micro (single molecule mass & speed) and macro worlds (kinetic energy released).