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Ludwig Boltzmann killed himself so that we could do thermodynamics by enumerating the states of a system. Learn to count, the most important skill in statistical mechanics.

A system consists of \( 4 \) coins. Each coin has a 50-50 probability to show heads or tails. Find the entropy of the case where all coins show heads.

The Boltzmann constant is \( k_B. \)

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We isothermally compress \( 7 \) mole of \( O_2 \) from \( 6 \text{ L} \) to \( 3 \text{ L}. \) How much does the entropy of the gas change?

**Assumptions and Details**

- \( O_2 \) is ideal gas
- \( N_A \) is Avogadro's number.

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\( 4 \) particles are in a room which is divided into two equal parts. If we consider the macrostate of the even split, what is its entropy?

**Assumptions and Details**

- The room temperature is \( 33 ^\circ \text{C}. \)
- The Boltzmann constant is \( k_B\)

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A \( 6 \text{ L} \) container is separated by a partition into two equal halves. Each half contains \( 8 \text{ mole} \) of either \(\text{He}\) or \(\text{Xe}\). What is the change in entropy of the total system after the partition is removed?

**Assumptions and Details**

- \(\text{He}\), and \(\text{Xe}\) are ideal gases.
- \( N_A \) is Avogadro's number.

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