Question

What volume does \(0.103 \mathrm{mol}\) of \(\mathrm{N}_{2}\) gas occupy at a temperature of \(27^{\circ} \mathrm{C}\) and a pressure of \(784 \mathrm{mm} \mathrm{Hg} ?\)

Short answer

The volume of \(0.103 \mathrm{mol}\) of \(\mathrm{N}_{2}\) gas at a temperature of \(27^{\circ} \mathrm{C}\) and a pressure of \(784 \mathrm{mm} \mathrm{Hg}\) is approximately \(2.52 \mathrm{L}\).

Step-by-step solution

Convert given values to appropriate units

First, let's convert the given temperature from Celsius to Kelvin and the given pressure from mm Hg to atm. To convert the temperature to Kelvin, add 273.15 to the Celsius value: Temperature in Kelvin (T) = 27°C + 273.15 = 300.15 K To convert the pressure to atm, divide the mm Hg value by 760, as there are 760 mm Hg in 1 atm: Pressure in atm (P) = 784 mm Hg / 760 = 1.0316 atm

Use the Ideal Gas Law

Now, we will use the Ideal Gas Law (PV = nRT) to find the volume occupied by the gas. We already have the pressure (P), number of moles (n), and temperature (T) in appropriate units. So, let's plug in those values, along with the Ideal Gas Constant (R) in atm·L/mol·K, which is 0.0821: 1.0316 atm · V = 0.103 mol · 0.0821 atm·L/mol·K · 300.15 K

Solve for the volume (V)

Now, we just need to isolate the volume (V) on one side of the equation and solve for it: V = (0.103 mol · 0.0821 atm·L/mol·K · 300.15 K) / 1.0316 atm V = 2.515067 L

Report the final answer

Now that we have found the volume, let's report the answer with the appropriate unit: The volume of 0.103 mol of N₂ gas at a temperature of 27°C and a pressure of 784 mm Hg is approximately 2.52 L.

Key concepts

Gas Volume Calculation

Have you ever wondered how we can determine the space a gas occupies? It's all about understanding the gas volume calculation. When calculating the volume of a gas, we use the Ideal Gas Law. This is a fundamental principle in chemistry which states:\[ PV = nRT \]
Here:

• \( P \) is the pressure of the gas.
• \( V \) is the volume of the gas.
• \( n \) is the amount of substance of the gas (in moles).
• \( R \) is the ideal gas constant, which is approximately \(0.0821 \text{ atm} \cdot \text{L/mol} \cdot \text{K}\).
• \( T \) is the temperature of the gas in Kelvin.

The Ideal Gas Law helps us calculate the unknown volume if all other variables are known. For instance, knowing the pressure, temperature, and number of moles lets you rearrange the formula to solve for volume. That's how you find that certain amount of gas occupies a specific space under given conditions.
If you remember this formula, solving volume-related problems becomes straightforward. Just ensure to convert your units correctly before using the formula.

Unit Conversion

Unit conversion is a crucial step in solving gas volume calculations accurately. We often deal with different units for pressure and temperature, which can be confusing at first. Let's take temperature and pressure as examples:

• Temperature: In gas calculations, temperature needs to be in Kelvin. To convert from Celsius to Kelvin, simply add 273.15 to your Celsius temperature. For example, \(27^{\circ} \mathrm{C}\) converts to \(300.15 \mathrm{K}\).


• Pressure: Pressure units like mm Hg (also known as Torr) are often measured when dealing with gas laws. However, it's best converted to the standard atmospheric pressure unit (atm) for the Ideal Gas Law. Since \(760 \text{ mm Hg} = 1 \text{ atm}\), you would divide your mm Hg measurement by 760 to convert it to atm. For example, \(784 \text{ mm Hg}\) is approximately \(1.0316 \text{ atm}\).

By converting all your units correctly, you ensure accuracy in your calculations and avoid potential errors. It may seem like an extra step, but it's vital for the right result. Once the units are in check, you can confidently use the Ideal Gas Law.

Pressure and Temperature in Gases

The study of gases involves understanding how pressure and temperature affect them. These two variables have a significant impact on the behavior and volume of a gas. Let's explore how they interact. When it comes to gases:

• Pressure: This is the force that the gas exerts on the walls of its container. It's measured in units like atm or mm Hg. The higher the pressure, the more the gas molecules are forced into a space, typically decreasing the volume, according to Boyle's Law. This is because gas molecules are more compressed under high pressure.


• Temperature: Temperature is a measure of the kinetic energy of gas molecules. Measured in Kelvin for calculations, it's directly proportional to volume in the Ideal Gas Law, described by Charles’s Law. This means that as temperature increases, so does the volume, if pressure is constant. Gas molecules gain energy with heat, making them move more and occupy more space.

By understanding these properties, you can predict and calculate changes in gas systems effectively. It's amazing how these fundamental concepts allow us to harness the gases around us! Always remember, a change in either pressure or temperature requires adjusting the other to maintain balance in a closed system.