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Chapter 20: Problem 17

Prove that Boltzmann's microscopic definition of entropy, \(S=k_{\mathrm{B}} \ln w\), implies that entropy is an additive variable: Given two systems, A and B, in specified thermodynamic states, with entropies \(S_{A}\) and \(S_{\mathrm{p}}\), respectively, show that the corresponding entropy of the combined system is \(S_{\mathrm{A}}+S_{\mathrm{B}}\).

Short Answer

Expert verified
Question: Prove that entropy is an additive variable using Boltzmann's microscopic definition of entropy. Answer: Using the Boltzmann formula, \(S = k_B \ln w\), we expressed the entropies of two independent systems A and B, and their combined system (A+B). By determining the number of microstates for the combined system (A+B) as the product of the individual systems' microstates, we were able to demonstrate that the entropy of the combined system (A+B) is equal to the sum of the entropies of the individual systems, \(S_A + S_B\), which proves that entropy is an additive variable.

Step by step solution

01

Write down the Boltzmann formula for System A and System B

We first write down the Boltzmann formula for System A and System B. This will help us set up the given entropy expressions for both systems. \[ S_A=k_B \ln w_A, \quad S_B = k_B \ln w_B \]
02

Express the number of microstates for the combined system

Now, we need to determine the number of microstates for the combined System A+B. Since the systems are independent, the total number of microstates for the combined system is equal to the product of the individual systems: \[ w_{A+B} = w_A \cdot w_B \]
03

Write down the Boltzmann formula for the combined system

Using the Boltzmann formula \(S = k_B \ln w\), we can express the entropy for the combined system as follows: \[ S_{A+B} = k_B \ln w_{A+B} \] Now we can substitute the expression for the number of microstates of the combined system: \[ S_{A+B} = k_B \ln(w_A \cdot w_B) \]
04

Use logarithm properties to split the sum

Next, we take advantage of the logarithm properties to separate the sum of the individual entropies. Recall that \(\ln(ab) = \ln(a) + \ln(b)\), so the entropy of the combined system can be expressed as the sum of logarithms: \[ S_{A+B} = k_B (\ln w_A + \ln w_B) \]
05

Show that the entropy of the combined system is the sum of the entropies of the individual systems

Now, we simply need to recognize that the result obtained in Step 4 can be rewritten in terms of the entropies of the individual systems: \[ S_{A+B} = k_B \ln w_A + k_B \ln w_B = S_A + S_B \] This proves that the entropy of the combined system is equal to the sum of the entropies of the individual systems, demonstrating that entropy is an additive variable.

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Most popular questions from this chapter

A certain refrigerator is rated as being \(32.0 \%\) as ef ficient as a Carnot refrigerator. To remove \(100 .\) J of heat from the interior at \(0^{\circ} \mathrm{C}\) and eject it to the outside at \(22^{\circ} \mathrm{C}\), how much work must the refrigerator motor do?

A proposal is submitted for a novel engine that will operate between \(400 . \mathrm{K}\) and \(300 . \mathrm{K}\) a) What is the theoretical maximum efficiency of the engine? b) What is the total entropy change per cycle if the engine operates at maximum efficiency?

A heat pump has a coefficient of performance of 5.0. If the heat pump absorbs 40.0 cal of heat from the cold outdoors in each cycle, what is the amount of heat expelled to the warm indoors?

Suppose an atom of volume \(V_{\mathrm{A}}\) is inside a container of volume \(V\). The atom can occupy any position within this volume. For this simple model, the number of states available to the atom is given by \(V / V_{A}\). Now suppose the same atom is inside a container of volume \(2 V .\) What will be the change in entropy?

A heat engine consists of a heat source that causes a monatomic gas to expand, pushing against a piston, thereby doing work. The gas begins at a pressure of \(300 . \mathrm{kPa}\), a volume of \(150 . \mathrm{cm}^{3}\), and room temperature, \(20.0^{\circ} \mathrm{C}\). On reaching a volume of \(450 . \mathrm{cm}^{3}\), the piston is locked in place, and the heat source is removed. At this point, the gas cools back to room temperature. Finally, the piston is unlocked and used to isothermally compress the gas back to its initial state. a) Sketch the cycle on a \(p V\) -diagram. b) Determine the work done on the gas and the heat flow out of the gas in each part of the cycle. c) Using the results of part (b), determine the efficiency of the engine.
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