Question

An open vessel having air is heated from \(27^{\circ} \mathrm{C}\) to \(127^{\circ} \mathrm{C}\). The fraction of air which goes out with respect to originally present is: (a) \(2 / 3\) (b) \(1 / 3\) (c) \(3 / 4\) (d) \(1 / 4\)

Short answer

(d) \(\frac{1}{4}\)

Step-by-step solution

Understand the relationship between temperature and volume

The problem involves heating the air in an open vessel. According to Charles's Law, for constant pressure, the volume of a gas is directly proportional to its absolute temperature. This means as the temperature increases, the volume of the gas expands.

Convert temperatures from Celsius to Kelvin

To apply Charles's Law, we need to work with temperatures in Kelvin. Convert the initial and final temperatures:- Initial temperature: \(27^{\circ} \text{C} = 27 + 273 = 300 \text{ K}\)- Final temperature: \(127^{\circ} \text{C} = 127 + 273 = 400 \text{ K}\)

Apply Charles's Law to find the fraction of air

According to Charles's Law:\[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]In this open vessel, the volume \(V_2\) will become equal to the initial volume plus the volume of air that has left. So, simplifying, the fraction of volume that remains is the ratio of initial to final temperatures:\[ \frac{V_1}{V_2} = \frac{T_1}{T_2} = \frac{300}{400} = \frac{3}{4} \]The remaining fraction of the air is \(\frac{3}{4}\), so the fraction that leaves is \(1 - \frac{3}{4} = \frac{1}{4}\).

Key concepts

Gas Expansion

When air or any gas is heated, it expands. This means that the particles of the gas move faster and tend to occupy more space. This phenomenon is known as gas expansion. In an open vessel, as you heat the air, the gas spreads out and starts to escape from the container. The expansion occurs because the gas particles gain energy and move further apart.

• Hotter gas particles move faster.
• Faster movement leads to more space being occupied.
• In an open system, such as an open vessel, gas expansion leads to some of the gas particles leaving the vessel.

The expansion of gas is a fundamental concept in understanding how temperature changes affect gases, especially when they are not confined or restricted.

Temperature Conversion

Temperature conversion is a key process when dealing with gas laws, such as Charles's Law. Gas laws work with the Kelvin scale, which is an absolute temperature scale based on absolute zero. To convert temperatures from Celsius to Kelvin, simply add 273. This addition aligns the Celsius scale with an absolute scale where zero is the lowest possible temperature.

• Celsius to Kelvin: Add 273.
• Kelvin scale is used because it starts at absolute zero.
• Allows for a direct relationship in gas calculations.

For example, converting 27°C to Kelvin gives 300 K, and 127°C converts to 400 K. Remembering to convert temperatures correctly allows accurate application of gas laws like Charles's Law.

Absolute Temperature

The concept of absolute temperature is crucial for understanding gas laws. Absolute temperature is measured on the Kelvin scale, which begins at absolute zero. Absolute zero is the point at which molecular motion stops, meaning there is no thermal energy.

• Absolute zero is the starting point of the Kelvin scale.
• Gas laws require temperatures to be in Kelvin to ensure direct proportionality.
• Helps in accurately calculating relationships involving temperature and volume or pressure.

When using absolute temperature, you ensure that any proportional relationships in gas laws are maintained. This is why converting temperatures to Kelvin is not just a formality, but a necessity, for correct calculations.

Volume and Temperature Relationship

Charles's Law describes the direct relationship between the volume of a gas and its absolute temperature when pressure is held constant. This means if you increase the temperature , the volume increases proportionally. Conversely, when the temperature decreases, so does the volume.

• Volume \(\propto\) Absolute Temperature.
• Known formula: \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]
• Only valid for constant pressure conditions.

In an open container, as the temperature rises from 300 K to 400 K, the air expands and some of it escapes. Initially, the volume of air matches the volume of the vessel. When heated, the new volume is greater, leading to some air leaving the vessel so the remaining air adjusts to the new temperature. The formula helps calculate what proportion of air remains or leaves due to expansion.