A balloon is filled with helium gas with a partial pressure of 1.00 atm and neon gas with a partial pressure of 0.50 atm. For each of the following changes (a to e) of the initial balloon, select the diagram (A, B, or C) that shows the final volume of the balloon:
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d. The Kelvin temperature doubles and half of the gas atoms leak out (P does not change).

Verified step by step guidance

1. Step 1: Begin by identifying the relationship between the variables in the problem. The problem involves changes in temperature and the number of gas atoms, while pressure remains constant. Use the combined gas law: \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is temperature.
2. Step 2: Analyze the changes described in the problem. The Kelvin temperature doubles, so \( T_{final} = 2T_{initial} \). Half of the gas atoms leak out, meaning \( n_{final} = \frac{1}{2}n_{initial} \). Since pressure \( P \) does not change, the relationship between \( V \), \( n \), and \( T \) can be expressed as \( V \propto \frac{nT}{P} \).
3. Step 3: Substitute the changes into the proportionality equation. Initially, \( V_{initial} \propto n_{initial}T_{initial} \). After the changes, \( V_{final} \propto \frac{1}{2}n_{initial} \cdot 2T_{initial} \). Simplify this expression to find \( V_{final} \propto n_{initial}T_{initial} \), which indicates that the final volume is equal to the initial volume.
4. Step 4: Interpret the diagram provided. The balloon's volume should remain unchanged because the doubling of temperature compensates for the halving of the number of gas atoms, keeping the product \( nT \) constant.
5. Step 5: Select the diagram that matches the unchanged volume of the balloon. Based on the analysis, the correct diagram is the one that shows the balloon with the same volume as the initial state.

Key Concepts

Partial Pressure

Partial pressure refers to the pressure exerted by a single type of gas in a mixture of gases. In this scenario, the balloon contains helium and neon gases, each contributing to the total pressure. Understanding partial pressure is crucial for predicting how changes in gas composition or conditions affect the overall behavior of the gas mixture.

Ideal Gas Law

The Ideal Gas Law (PV=nRT) relates the pressure (P), volume (V), temperature (T), and number of moles (n) of a gas. In this question, the doubling of the Kelvin temperature while considering the number of gas atoms that leak out directly impacts the volume of the balloon. This law helps in understanding how gas behavior changes with temperature and quantity.

Gas Leakage and Volume Change

When gas leaks from a balloon, the number of gas particles decreases, which typically leads to a reduction in pressure and volume. However, in this case, the pressure remains constant while the temperature increases, which complicates the expected outcome. Analyzing how these factors interplay is essential for determining the final volume of the balloon after the specified changes.