Question

An 11.2-L sample of gas is determined to contain \(0.50\) mole of \(\mathrm{N}_{2}\). At the same temperature and pressure, how many moles of gas would there be in a 20.-L sample?

Short answer

The number of moles in the 20 L sample can be determined using the proportionality relationship between volume and moles of a gas. Given a 11.2 L sample containing 0.50 moles of N2, we can calculate the number of moles in the 20 L sample as: \[n_2 = \frac{n_1}{V_1} \times V_2\] By substituting the given values and performing the calculations, we find that the 20 L sample contains approximately 0.89 moles of gas (rounded to 2 decimal places).

Step-by-step solution

Identify the proportionality relationship between volume and moles of a gas

Under the same temperature and pressure conditions, the volume of a gas is directly proportional to its moles. We can use this relationship to find the number of moles in the different volume. Let's denote the given values as follows: - \(V_1 = 11.2 \,\text{L}\) (volume of the first sample) - \(n_1 = 0.50\,\text{mole}\) (moles of N2 in the first sample) - \(V_2 = 20\,\text{L}\) (volume of the second sample) The relationship between the two samples can be written as: \[\frac{n_1}{V_1} = \frac{n_2}{V_2}\] Where \(n_2\) is the number of moles in the 20 L sample, which is the value we need to find.

Rearrange the equation for the unknown value

We need to isolate \(n_2\) in the proportionality equation above. Rearrange the equation to solve for the unknown value, which is \(n_2\): \[n_2 = \frac{n_1}{V_1} \times V_2\]

Substitute the given values in the equation

Now that the equation is ready, substitute the given values for the volume and moles of the first sample and the volume of the second sample: \[n_2 = \frac{0.50\,\text{mole}}{11.2\,\text{L}} \times 20\,\text{L}\]

Calculate the number of moles in the 20 L sample

Perform the calculations to find the value of \(n_2\): \[n_2 = \frac{0.50\,\text{mole}}{11.2\,\text{L}} \times 20\,\text{L} = 0.8929\,\text{mole}\] The number of moles in the 20 L sample is approximately 0.89 moles (rounded to 2 decimal places).

Key concepts

Volume-to-mole relationship

The volume-to-mole relationship is a fundamental concept in the study of gases. It is derived from the Ideal Gas Law, which states that for a given amount of gas at constant temperature and pressure, the volume of the gas is directly proportional to the number of moles. This means that if you know the volume of a gas, you can determine its moles, and vice versa.

• If the volume of gas increases, the number of moles increases as long as temperature and pressure remain constant.
• Conversely, if the volume decreases, the number of moles decreases under the same conditions.

This relationship is powerful because it allows us to predict how changes in volume affect the quantity of gas. This prediction is based solely on proportional changes between volume and moles.

Proportionality in gases

Proportionality in gases focuses on how various properties such as volume, pressure, and temperature interact with the quantity of gas. For gases, under conditions where temperature and pressure are constant, volume is directly proportional to the number of moles.
To understand this better, think of gases as collections of tiny particles. When you have more particles (or moles), they take up more space, thus increasing volume. This proportional relationship can be expressed mathematically as:
\[\frac{n_1}{V_1} = \frac{n_2}{V_2}\]

• This equation shows that the ratio of moles to volume is consistent across different samples of a gas.
• This invariant ratio enables scientists and students to find unknown quantities based on changes in volume or moles.

Understanding this proportionality allows for practical calculations in laboratory settings or everyday situations involving gases.

Moles calculation

Moles calculation is a critical step in solving problems involving gases. By using known quantities and the relationships defined by the Ideal Gas Law, you can solve for unknowns effectively. In our context, we are applying the volume-to-mole proportionality.
Here is how you typically approach a moles calculation:1. **Identify known values:** Start by determining what you already know — such as initial volume, initial moles, and new volume. 2. **Set up the relationship:** Use the proportional relationship \(\frac{n_1}{V_1} = \frac{n_2}{V_2}\) to set up your calculation.3. **Solve for the unknown:** Rearrange the formula to solve for the unknown quantity, which is usually either moles or volume.

• Make sure each unit of measure is in consistent units to avoid errors.
• Plug in the numbers correctly to find the solution.

This approach to moles calculation helps demystify seemingly complex problems and provides a clear pathway from known to unknown values.