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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

Suppose a Brayton engine (see Problem 20.26 ) is run as a refrigerator. In this case, the cycle begins at temperature \(T_{1}\), and the gas is isobarically expanded until it reaches temperature \(T_{4}\). Then the gas is adiabatically compressed, until its temperature is \(T_{3}\). It is then isobarically compressed, and the temperature changes to \(T_{2}\). Finally, it is adiabatically expanded until it returns to temperature \(T_{1}\) - a) Sketch this cycle on a \(p V\) -diagram. b) Show that the coefficient of performance of the engine is given by \(K=\left(T_{4}-T_{1}\right) /\left(T_{3}-T_{2}-T_{4}+T_{1}\right) .\)

A coal-burning power plant produces \(3000 .\) MW of thermal energy, which is used to boil water and produce supersaturated steam at \(300 .{ }^{\circ} \mathrm{C}\). This high-pressure steam turns a turbine producing \(1000 .\) MW of electrical power. At the end of the process, the steam is cooled to \(30.0^{\circ} \mathrm{C}\) and recycled. a) What is the maximum possible efficiency of the plant? b) What is the actual efficiency of the plant? c) To cool the steam, river water runs through a condenser at a rate of \(4.00 \cdot 10^{7} \mathrm{gal} / \mathrm{h}\). If the water enters the condenser at \(20.0^{\circ} \mathrm{C}\), what is its exit temperature?

A refrigerator with a coefficient of performance of 3.80 is used to \(\operatorname{cool} 2.00 \mathrm{~L}\) of mineral water from room temperature \(\left(25.0^{\circ} \mathrm{C}\right)\) to \(4.00^{\circ} \mathrm{C} .\) If the refrigerator uses \(480 . \mathrm{W}\) how long will it take the water to reach \(4.00^{\circ} \mathrm{C}\) ? Recall that the heat capacity of water is \(4.19 \mathrm{~kJ} /(\mathrm{kg} \mathrm{K}),\) and the density of water is \(1.00 \mathrm{~g} / \mathrm{cm}^{3}\). Assume the other contents of the refrig. erator are already at \(4.00^{\circ} \mathrm{C}\).

Which of the following processes always results in an increase in the energy of a system? a) The system loses heat and does work on the surroundings. b) The system gains heat and does work on the surroundings. c) The system loses heat and has work done on it by the surroundings. d) The system gains heat and has work done on it by the surroundings. e) None of the above.

The change in entropy of a system can be calculated because a) it depends only on the c) entropy always increases. initial and final states. d) none of the above. b) any process is reversible.
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