CALC A lonely party balloon with a volume of 2.40 L and containing 0.100 mol of air is left behind to drift in the temporarily uninhabited and depressurized International Space Station. Sunlight coming through a porthole heats and explodes the balloon, causing the air in it to undergo a free expansion into the empty station, whose total volume is 425 m^3. Calculate the entropy change of the air during the expansion.

Verified step by step guidance

1. Identify the initial and final states of the air. Initially, the air is confined to a volume of 2.40 L (which needs to be converted to cubic meters for consistency in units). Finally, the air expands freely into the space station with a volume of 425 m^3.
2. Use the formula for the entropy change for an ideal gas undergoing a free expansion: \(\Delta S = nR \ln\left(\frac{V_f}{V_i}\right)\), where \(n\) is the number of moles of the gas, \(R\) is the ideal gas constant (8.314 J/mol\cdot K), \(V_i\) is the initial volume, and \(V_f\) is the final volume.
3. Convert the initial volume from liters to cubic meters by using the conversion factor 1 L = 0.001 m^3. Calculate \(V_i\) in cubic meters.
4. Substitute the values of \(n\), \(R\), \(V_i\), and \(V_f\) into the entropy change formula to set up the calculation.
5. Calculate the natural logarithm of the ratio \(\frac{V_f}{V_i}\) and then multiply by \(nR\) to find the entropy change \(\Delta S\).

Key Concepts

Entropy

Entropy is a measure of the disorder or randomness in a system. In thermodynamics, it quantifies the amount of energy in a physical system that is not available to do work. When a gas expands freely, such as in the case of the balloon bursting in a vacuum, the entropy of the gas increases because the molecules have more available microstates to occupy, leading to greater disorder.

Free Expansion

Free expansion refers to the process where a gas expands into a vacuum without doing work on the surroundings and without heat exchange. In this scenario, the gas expands spontaneously due to the absence of external pressure. This process is irreversible and results in an increase in entropy, as the gas molecules spread out to occupy a larger volume.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and number of moles of an ideal gas. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature. This law helps in calculating various properties of gases, including changes in state during processes like expansion.