Problem 35 - Chemistry in 30 days

Question

If an open vessel is heated from \(27^{\circ} \mathrm{C}\) to \(627^{\circ} \mathrm{C}\), what fraction of the molecules of air is left in the vessel ? (a) \(2 / 3\) (b) \(1 / 3\) (c) \(1 / 4\) (d) \(3 / 4\)

Step-by-Step Solution

Answer

The fraction of molecules left is \(1/3\) (option b).

Step 1: Understand the Problem

We have to determine what fraction of air molecules remain when the temperature changes from \(27^{\circ} \mathrm{C}\) to \(627^{\circ} \mathrm{C}\). The vessel is open, allowing some molecules to leave. It's essential to apply the principles of gas laws, specifically the ideal gas law and Charles's law, which considers volume and temperature changes.

Step 2: Convert Temperatures to Kelvin

Temperature needs to be in Kelvin for gas law calculations. Convert \(27^{\circ} \mathrm{C}\) and \(627^{\circ} \mathrm{C}\) to Kelvin: \(T_1 = 27 + 273 = 300\, \mathrm{K}\) and \(T_2 = 627 + 273 = 900\, \mathrm{K}\).

Step 3: Use Charles's Law to Find the Fraction of Molecules Remaining

According to Charles's Law, \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\). Since the vessel is open, the volume \(V\) remains unchanged, and we're interested in the mole fraction instead. Thus: \(\frac{n_2}{n_1} = \frac{T_1}{T_2}\).

Step 4: Calculate the Fraction Using Temperatures

Substitute the temperatures into the equation: \(\frac{n_2}{n_1} = \frac{300}{900}\). Simplifying this gives \(\frac{n_2}{n_1} = \frac{1}{3}\).

Step 5: Conclude the Result

Thus, the fraction of air molecules left in the vessel when it is heated from \(27^{\circ} \mathrm{C}\) to \(627^{\circ} \mathrm{C}\) is \(\frac{1}{3}\). The correct answer is (b) \(1/3\).

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a crucial formula in chemistry and physics that describes the state of an ideal gas. It intertwines the four important properties of gases: pressure (P), volume (V), temperature (T), and number of moles (n). This relationship is mathematically expressed as \( PV = nRT \), where \( R \) is the ideal gas constant.

The Ideal Gas Law assumes that:

• The gas particles do not attract or repel each other.
• The volume of the individual gas particles is negligible.
• The energy is distributed evenly among all particles.

This law is widely applicable for approximating the behavior of gases under many conditions. However, it's important to note that at high pressures or very low temperatures, real gases deviate from this ideal behavior. To solve problems concerning changing conditions of a gas, such as those in open or closed systems, it's essential to understand how the Ideal Gas Law helps link moles, volume, and temperature.

Charles's Law

Charles's Law is an important concept that explains the direct relationship between the volume and temperature of a gas, provided the pressure and number of moles remain constant. It states that as the temperature of a gas increases, its volume increases as well, as long as the pressure remains unchanged.

Mathematically, Charles's Law is expressed as \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \), where \( V \) is volume and \( T \) is temperature (in Kelvin). This shows that when temperature rises, the gas expands if allowed to do so.

In the given problem, we used Charles's Law to determine how much air remains in the vessel as it heats up. When the vessel is open, it allows air particles to escape, adjusting the density of the air within according to the increased temperature. This ultimately leads to an important conclusion, which was calculated using the law: only one-third of the original air molecules remain under the new conditions.

Mole Fraction

Mole Fraction is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of the number of moles of one component to the total number of moles in the mixture. The mole fraction, often denoted by \( X \), is a dimensionless quantity as it represents a simple ratio.

The formula to calculate the mole fraction of a component A in a mixture is \( X_A = \frac{n_A}{n_{total}} \), where \( n_A \) is the number of moles of A, and \( n_{total} \) is the total number of moles of all components in the mixture.

Using concepts like mole fraction, we can accurately describe changes in a system’s composition. In the context of the heated open vessel, calculating the remaining mole fraction of air molecules enhances our understanding of how heat affects gaseous mixtures. As temperature increases, molecules exit the vessel, thus reducing the total number of moles and affecting the mole fraction of the remaining gas.