Question

What size container do you need to hold 0.0459 mol of \(N_{2}\) gas at STP?

Short answer

The volume of the container needed to hold 0.0459 mol of N₂ gas at STP is approximately 1.029 L.

Step-by-step solution

Identify given values and constants

Given: Moles of N₂ gas (n) = 0.0459 mol Standard Pressure (P) = 1 atm Standard Temperature (T) = 273.15 K Ideal Gas Constant (R) = 0.0821 Latm/molK

Rearrange the Ideal Gas Law equation to solve for Volume (V)

Since we need to find the volume, we'll rearrange the Ideal Gas Law equation to solve for V. \[V = \frac{nRT}{P}\]

Plug in the given values and solve for Volume (V)

Next, we substitute the given values and constants into the equation and solve for V: \[V = \frac{(0.0459\ mol)(0.0821\ \frac{L\cdot atm}{mol\cdot K})(273.15\ K)}{1\ atm}\]

Calculate Volume (V)

Now, we perform the calculations: \[V = \frac{(0.0459)(0.0821)(273.15)}{1}\] \[V \approx 1.029\ L\]

Write the answer in a proper form

The volume of the container needed to hold 0.0459 mol of N₂ gas at STP is approximately 1.029 L.

Key concepts

Molar Volume of Gas

The molar volume of a gas is the volume occupied by one mole of the gas under specified conditions. At STP (standard temperature and pressure), which is 0°C (273.15 K) and 1 atmosphere of pressure, one mole of any ideal gas occupies a volume of 22.4 liters. This is a convenient figure to remember for solving many gas-related problems.

Understanding molar volume is crucial when you need to determine how much space a known amount of gas will occupy at STP. For example, in the problem provided, calculating the volume needed for 0.0459 mol of nitrogen gas involves the use of molar volume under these standard conditions.

STP Conditions

STP stands for Standard Temperature and Pressure. It is a common reference for gases, defined as a temperature of 0°C (273.15 K) and a pressure of 1 atmosphere (atm). STP allows scientists and engineers to use the ideal gas law and compare the behavior of gases. All gases at STP have the same molar volume, which simplifies calculations and comparisons.

When working with gases, it's essential to know if the STP conditions are being used, as this will determine the variables used in equations like the ideal gas law. In our exercise, we use STP conditions to find the volume required for a known amount of gas.

Gas Constant

The gas constant, denoted as R, is a proportionality factor in the ideal gas law, which relates the volume, temperature, and pressure of a mole of gas. Its value depends on the units used for pressure, volume, and temperature. In the context of STP and using liters, atmospheres, and kelvins, the value of 'R' is 0.0821 L·atm/mol·K.

The gas constant is essential in gas law calculations as it bridges the elements of the ideal gas equation, allowing us to solve for any unknown variable provided the others are known. This value was used in the exercise to calculate the volume of the gas at STP.

Moles to Volume Calculation

To convert moles of a gas to volume, we use the ideal gas law equation, which mathematically relates the amount of substance (in moles), the temperature (T), the pressure (P), and the volume (V). The rearranged form of the equation to solve for the volume of a gas at STP is:
\[ V = \frac{nRT}{P} \]
Where:

• \( V \) is the volume of the gas,
• \( n \) is the number of moles,
• \( R \) is the gas constant, and
• \( P \) is the pressure of the gas.

By inserting the values of moles, standard temperature, and pressure into this formula, you can calculate the exact volume the gas will occupy at STP, just as it was demonstrated in the solved example.