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title: "Four distinguishable gas particles are initially confined to bulb A of a two-bulb apparatus consisting of identical bulbs A and B connected by a thin tube with a closed valve. The valve is then opened, allowing the particles to move freely between the bulbs. Which of the following statements correctly compares the probability \\(P_{(4,0)}\\) of finding all four particles in bulb A to the probability \\(P_{(2,2)}\\) of finding two particles in each bulb, and provides a correct explanation based on statistical mechanics?"
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url: "https://nerd-notes.com/ubq/116255/"
date_modified: "2026-08-03T11:46:28+00:00"
---

# Four distinguishable gas particles are initially confined to bulb A of a two-bulb apparatus consisting of identical bulbs A and B connected by a thin tube with a closed valve. The valve is then opened, allowing the particles to move freely between the bulbs. Which of the following statements correctly compares the probability \(P_{(4,0)}\) of finding all four particles in bulb A to the probability \(P_{(2,2)}\) of finding two particles in each bulb, and provides a correct explanation based on statistical mechanics?

Four distinguishable gas particles are initially confined to bulb A of a two-bulb apparatus consisting of identical bulbs A and B connected by a thin tube with a closed valve. The valve is then opened, allowing the particles to move freely between the bulbs. Which of the following statements correctly compares the probability \(P_{(4,0)}\) of finding all four particles in bulb A to the probability \(P_{(2,2)}\) of finding two particles in each bulb, and provides a correct explanation based on statistical mechanics?

![Two identical spherical bulbs, labeled Bulb A on the left and Bulb B on the right, are connected by a narrow horizontal tube with a valve in the middle. Inside Bulb A, four distinct circular particles labeled 1, 2, 3, and 4 are shown. Bulb B is completely empty. The valve on the connecting tube is shown in a closed position. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785757588-JUKL9X.jpg)

- **A.** \(P_{(4,0)}\) is \(\dfrac{1}{4} P_{(2,2)}\), because there are 4 total particles in the container and each particle has a 50% chance of being in bulb A.
- **B.** \(P_{(4,0)}\) is \(1 P_{(2,2)}\), because both configurations represent valid macroscopic equilibrium states with equal total thermal energy.
- **C.** \(P_{(4,0)}\) is \(\dfrac{1}{16} P_{(2,2)}\), because there are 16 total microstates for the system and state \((4,0)\) consists of only 1 microstate.
- **D.** \(P_{(4,0)}\) is \(\dfrac{1}{6} P_{(2,2)}\), because state \((2,2)\) corresponds to 6 microstates while state \((4,0)\) corresponds to 1 microstate, making state \((2,2)\) the state of maximum entropy.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/116255/*