Question

If \(23.2 \mathrm{g}\) of a given gas occupies a volume of \(93.2 \mathrm{L}\) at a particular temperature and pressure, what mass of the gas occupies a volume of \(10.4 \mathrm{L}\) under the same conditions?

Short answer

The mass of the gas that occupies a volume of 10.4 L under the same conditions is approximately \(2.58 \ \mathrm{g}\).

Step-by-step solution

Write down the proportion formula

A proportion relates two ratios by stating that they are equal. In this problem, when comparing the volumes and masses, we can set up a proportion as follows: \[ \frac{m_1}{V_1} = \frac{m_2}{V_2} \] where: - \(m_1\) is the mass of the first gas sample (23.2 g) - \(V_1\) is the volume of the first gas sample (93.2 L) - \(m_2\) is the unknown mass of the second gas sample - \(V_2\) is the volume of the second gas sample (10.4 L)

Plug in the given values

Now, let's substitute the given values into the proportion formula: \[ \frac{23.2 \ \mathrm{g}}{93.2 \ \mathrm{L}} = \frac{m_2}{10.4 \ \mathrm{L}} \]

Solve for the unknown mass, \(m_2\)

To solve for \(m_2\), we can simply cross multiply and solve for the unknown value: \[ m_2 = \frac{23.2 \ \mathrm{g} \times 10.4 \ \mathrm{L}}{93.2 \ \mathrm{L}} \] Now, calculate the value of \(m_2\): \[ m_2 \approx 2.5839 \ \mathrm{g} \]

Round the result to the correct significant figures

Since we have 3 significant figures in both given volumes (23.2 g, 93.2 L, and 10.4 L), we should round our final answer to 3 significant figures as well: \[ m_2 \approx 2.58 \ \mathrm{g} \] So, the mass of the gas that occupies a volume of 10.4 L under the same conditions is approximately 2.58 g.

Key concepts

Proportion

In mathematics, a proportion is an equation that states that two ratios are equivalent. Proportions help us understand the relationship between different quantities. In the context of gas laws, proportions allow us to compare the masses and volumes of gases under the same conditions.

For example, if 23.2 g of a gas occupies 93.2 L, and we want to find the mass of this gas that would occupy 10.4 L, we use a proportion. The general formula we use is:

• \( \frac{m_1}{V_1} = \frac{m_2}{V_2} \)

where:

• \( m_1 \) is the mass of the first gas sample (23.2 g).
• \( V_1 \) is the volume of the first gas sample (93.2 L).
• \( m_2 \) is the unknown mass of the second gas sample.
• \( V_2 \) is the volume of the second gas sample (10.4 L).

By rearranging and solving this equation, you can find \( m_2 \), which is the mass equivalent to the second volume. Remember, proportions are useful because they assume that the ratio between the variables remains constant under the same physical conditions.

Mass-Volume Relationship

The mass-volume relationship in gas calculations comes from the basic understanding of how gases occupy space in a container. Under constant temperature and pressure, the mass of gas in a given volume is directly proportional to that volume.

For the given problem:

• We have a known mass and volume: 23.2 g in 93.2 L.
• This relationship helps us determine how much mass fits into another specified volume of the same gas.

This is essential for calculations concerning gases because their particles fill the volume they occupy completely. The proportion we set up reflects this relationship, assuming ideal behavior in gases.

In practice, this direct relationship implies that cutting the volume will typically cut the mass by the corresponding ratio, making it simpler to predict how much of a substance will fit into a new volume given the same conditions.

Significant Figures

Understanding significant figures is crucial in scientific calculations as they reflect the precision of measurements. Significant figures are the digits in a number that contribute to its accuracy. This includes all non-zero digits, any zeros between significant digits, and any trailing zeros in a decimal number.

In gas calculations:

• The values given—23.2 g, 93.2 L, and 10.4 L—all have three significant figures.
• This consistency impacts the precision of your final answer. It is important to carry through the same level of precision in calculations.

In our calculation, we found \( m_2 \), initially calculated to be approximately 2.5839 g. However, to reflect the data's precision, we round this to three significant figures: 2.58 g.

Overall, maintaining the correct number of significant figures ensures the reliability of your result and matches the sensitivity provided by the measurement tools used to obtain the initial data.