Question 1

Consider a rectangular book. The book is (approximately) symmetric for rotations around an axis that goes diagonally from one corner to the other

Accepted Answer

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Question 2

Consider a system consisting of two Einstein solids $P$ and $Q$ in thermal contact. Assume that we know the number of atoms in each solid and $\varepsilon$. What do we know about the system if we also know the total energy in each of the two objects?

Accepted Answer

Knowing the total energy in each of the two objects, along with the number of atoms in each solid and $\varepsilon$, allows us to determine the macrostate of the system and how the total energy is divided between the two solids, which is the macropartition. However, it does not provide enough information to specify the exact microstate, as there are typically many microstates corresponding to a given macrostate.

Question 3

Consider a system consisting of two Einstein solids $P$ and $Q$ in thermal contact. Assume that we know the number of atoms in each solid and $\varepsilon$. What do we know about the system if we also know the total energy of the combined system?

Accepted Answer

Knowing the total energy of the combined system, along with the number of atoms in each solid and $\varepsilon$, allows us to determine the macrostate of the system. The macrostate is defined by macroscopic properties such as total energy, volume, and number of particles. However, this information does not specify the exact microstate (the specific quantum states of all particles) or how the energy is partitioned between the two solids (macropartition).

Question 4

What is the crucial characteristic of an Einstein solid that makes it easier to analyze in the context of this chapter than most other kinds of thermodynamic systems?

Accepted Answer

The crucial characteristic of an Einstein solid is that its microstates are comparatively easy to count, which simplifies the analysis of its thermodynamic properties.

Question 5

We can ignore an Einstein solid's zero-point energy because

Accepted Answer

The zero-point energy of an Einstein solid is a constant value that does not change during thermal interactions. This constancy means it does not affect calculations of energy differences, which are what matter in thermodynamics.

Question 6

Which of the following statements is true?

Accepted Answer

The answer of Which of the following statements is true?
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Question 7

Suppose that we increase an $N$-atom Einstein solid's value of $q=U / \varepsilon$ from $q$ to $q+1$. By what factor does the value of its multiplicity $\Omega$ increase? (Choose whichever result is the closest, assuming that $q>>1$ and $N>>1$.)

Accepted Answer

The multiplicity of an Einstein solid is given by $\Omega = \binom{q + N - 1}{q}$. For large $q$ and $N$, the ratio $\frac{\Omega(q+1)}{\Omega(q)}$ approximates $\frac{q+3N}{q}$.

Question 8

In the situation shown in figure T2.5c, the width of the bell curve at half its peak value is $0.021 U$. If we multiply $N_{A}$, $N_{B}$, and $U$ by a factor of 100 , then the peak's width (as you can check) is about $0.0021 U$. Suppose that we increase each solid's $N$ and the system's total energy by another factor of $10^{18}$ to create solids containing $1 / 6$ of a mole of atoms (still pretty small objects by everyday standards). Assuming that the trend continues, what will be the approximate width of the combined system's probability bell curve as a fraction of $U$ ?

Accepted Answer

The answer of In the situation shown in figure T2.5c,...

Question 9

Suppose that you use a super-fast computer to measure a system's macropartition a billion times a second. What is the approximate probability of the least-probable macropartition that you might plausibly see in your lifetime? Select the closest response.

Accepted Answer

If you measure a billion times a second, over a typical human lifetime (around 80 years), you would have approximately $10^{18}$ measurements. Thus, the least-probable macropartition you might see would have a probability of about 1 in $10^{18}$.

Question 10

Consider the system where $N_{A}=N_{B}=1000, U=5999 \varepsilon$ (see figure T2.6). Suppose that we can measure each object's energy to within $0.01 U \approx 60 \varepsilon$ (the size of one bin). Once the system has reached a macropartition in the most probable bin, it will never spontaneously move to a macropartition outside that bin.

Accepted Answer

Even in the most probable macropartition, there is always a non-zero probability, albeit very small, for the system to fluctuate to a less probable state due to the inherent statistical nature of thermodynamic systems. These fluctuations mean the system can spontaneously move to a macropartition outside the most probable bin, although such events are rare.

Question 11

Which of the systems listed below is the largest, having $N_{A}=N_{B}$ and $U=6 N_{A} \varepsilon$ for which there is a better than 1 in a billion chance of seeing one or the other solid having zero energy? (Hint: Using StatMech to check cases is probably the easiest approach.)

Accepted Answer

The correct answer is based on statistical mechanics principles, specifically the Boltzmann distribution and the concept of microstates. For a system with $N_{A}=N_{B}$ and $U=6N_{A}\varepsilon$, the probability of one solid having zero energy while the other has all the energy can be calculated using the number of microstates ($\Omega$) corresponding to these conditions. The probability of observing one solid at zero energy is significantly higher for smaller systems due to the exponential dependence on the system size in the Boltzmann factor. However, as $N_{A}$ increases, the number of microstates for distributing energy increases, making it increasingly unlikely to find all energy in one solid. The hint suggests using statistical mechanics to evaluate the likelihood for different $N_{A}$ values, and among the given options, $N_{A}=8$ is the largest system size for which there is still a better than 1 in a billion chance of observing one of the solids at zero energy, based on the calculation of probabilities from the distribution of energy states.

Question 12

A hot object is placed in contact with a cold object. We observe that heat flows spontaneously from the hot object to the cold object, but not in the other direction. According to the argument in this chapter, this is so because

Accepted Answer

The system evolves toward macropartitions with more microstates, which corresponds to an increase in the total entropy of the system, consistent with the second law of thermodynamics.

Question 13

Consider an object (object $A$ ) whose multiplicity is always 1 , no matter how much energy you put into it. If you put a very large amount of energy into such an object, and place it into thermal contact with an Einstein solid (object $B$ ) having the same number of atoms but much less energy, what will happen? (Hint: Imagine a macropartition table for such a pair of objects.)

Accepted Answer

Since object $A$ has a constant multiplicity of 1 regardless of its energy, it has no entropy change with energy changes. Object $B$, an Einstein solid, will have increasing multiplicity and entropy with energy. Therefore, energy will flow from $A$ to $B$ to increase the total entropy of the system, until $A$ has very little energy.

Question 14

In a given macropartition, the total entropy of a system comprised of two or more parts is always equal to the sum of the entropies of those parts.

Accepted Answer

The entropy of a system is a measure of its disorder. The more disordered a system is, the higher its entropy. When two or more systems are combined, the total entropy of the combined system is always greater than or equal to the sum of the entropies of the individual systems. This is because the combined system has more degrees of freedom than the individual systems, and therefore more ways to be disordered.

Question 15

$10^{62} / 10^{60}=100$. What is $\ln 10^{62} /$ In $10^{60}$ ? (Select the closest response.)

Accepted Answer

The logarithm of a quotient is the difference of the logarithms, so $\ln 10^{62} /\ln 10^{60} = \ln(10^{62}/10^{60}) = \ln 10^2 = \ln 100$. Since $\ln 100$ is approximately 4.6, the division as presented in the question is incorrect. The correct interpretation should be $\ln(10^{62}/10^{60}) = \ln 10^2 = \ln 100$, which simplifies to 2 times $\ln 10$, and since $\ln 10$ is approximately 2.3, the answer is approximately 4.6. However, given the options and the likely misinterpretation of the question, the closest correct response based on the intended operation (division of logarithms) is 1.0, because the question seems to mistakenly present division where it likely meant to describe a different operation.

Question 16

The entropy of a certain macropartition of a combined system is $102 \mathrm{k}_{\mathrm{B}}$. The entropy of another macropartition is $204 \mathrm{k}_{\mathrm{B}}$. How much more likely is the system to be in the second macropartition than the first? (Select the closest response.)

Accepted Answer

The probability of a macropartition is proportional to $e^{S}$, where $S$ is the entropy. So the ratio of probabilities is $e^{204 k_{B}}/e^{102 k_{B}} = e^{102 k_{B}} \approx 2 \times 10^{44}$.

Question 17

Which system has the greater entropy, a puddle of water sitting on a table next to a chaotic scattering of salt crystals or the same amount of salt dissolved in an evenly distributed way in the puddle? Be prepared to describe your reasoning.

Accepted Answer

The system with the salt dissolved in the puddle has greater entropy because the salt ions are more dispersed and there are more possible microstates for the system. When the salt is dissolved, the mixture is more homogeneous and the distribution of particles is more random, leading to higher entropy.

Question 18

Object A's entropy increases quite a bit when we give it a certain tiny amount of energy, whereas object B's entropy increases by a much smaller amount with the same gift. Which has the higher temperature?

Accepted Answer

The relationship between entropy ($S$) and temperature ($T$) for a given amount of energy ($\Delta Q$) is given by the formula $\Delta S = \frac{\Delta Q}{T}$. If object A's entropy increases significantly with a small amount of energy, it implies that its temperature is lower compared to object B, which shows a smaller increase in entropy for the same amount of energy. This is because the increase in entropy is inversely proportional to the temperature.

Question 19

As a normal object's thermal energy $U$ goes to zero, the slope of a graph of its entropy $S$ versus $U$

Accepted Answer

As thermal energy $U$ approaches zero, the temperature approaches absolute zero, and the slope $\frac{dS}{dU} = \frac{1}{T}$ approaches infinity because temperature $T$ approaches zero.

Question 20

Consider a system whose multiplicity is 1 , no matter how much energy one puts into it. What is such a system's temperature?

Accepted Answer

The temperature of a system is related to the change in entropy (or multiplicity) with respect to energy. If adding energy doesn't change the multiplicity (i.e., the multiplicity remains 1), the system's temperature is undefined because the formula for temperature involves a division by the change in entropy, which in this case is zero.

Quiz 6: Some Processes Are Irreversible Thermal Physics

Consider a rectangular book. The book is (approximately) symmetric for rotations around an axis that goes diagonally from one corner to the other

Consider a system consisting of two Einstein solids $P$ and $Q$ in thermal contact. Assume that we know the number of atoms in each solid and $\varepsilon$ . What do we know about the system if we also know the total energy in each of the two objects?

Consider a system consisting of two Einstein solids $P$ and $Q$ in thermal contact. Assume that we know the number of atoms in each solid and $\varepsilon$ . What do we know about the system if we also know the total energy of the combined system?

What is the crucial characteristic of an Einstein solid that makes it easier to analyze in the context of this chapter than most other kinds of thermodynamic systems?

We can ignore an Einstein solid's zero-point energy because

Which of the following statements is true?

Suppose that we increase an $N$ -atom Einstein solid's value of $q=U / \varepsilon$ from $q$ to $q+1$ . By what factor does the value of its multiplicity $\Omega$ increase? (Choose whichever result is the closest, assuming that $q>>1$ and $N>>1$ .)

Suppose that you use a super-fast computer to measure a system's macropartition a billion times a second. What is the approximate probability of the least-probable macropartition that you might plausibly see in your lifetime? Select the closest response.

Which of the systems listed below is the largest, having $N_{A}=N_{B}$ and $U=6 N_{A} \varepsilon$ for which there is a better than 1 in a billion chance of seeing one or the other solid having zero energy? (Hint: Using StatMech to check cases is probably the easiest approach.)

A hot object is placed in contact with a cold object. We observe that heat flows spontaneously from the hot object to the cold object, but not in the other direction. According to the argument in this chapter, this is so because

In a given macropartition, the total entropy of a system comprised of two or more parts is always equal to the sum of the entropies of those parts.

$10^{62} / 10^{60}=100$ . What is $\ln 10^{62} /$ In $10^{60}$ ? (Select the closest response.)

The entropy of a certain macropartition of a combined system is $102 \mathrm{k}_{\mathrm{B}}$ . The entropy of another macropartition is $204 \mathrm{k}_{\mathrm{B}}$ . How much more likely is the system to be in the second macropartition than the first? (Select the closest response.)

Which system has the greater entropy, a puddle of water sitting on a table next to a chaotic scattering of salt crystals or the same amount of salt dissolved in an evenly distributed way in the puddle? Be prepared to describe your reasoning.

Object A's entropy increases quite a bit when we give it a certain tiny amount of energy, whereas object B's entropy increases by a much smaller amount with the same gift. Which has the higher temperature?

As a normal object's thermal energy $U$ goes to zero, the slope of a graph of its entropy $S$ versus $U$

Consider a system whose multiplicity is 1 , no matter how much energy one puts into it. What is such a system's temperature?

Quiz 6: Some Processes Are Irreversible Thermal Physics

Consider a rectangular book. The book is (approximately) symmetric for rotations around an axis that goes diagonally from one corner to the other

Consider a system consisting of two Einstein solids PPP and QQQ in thermal contact. Assume that we know the number of atoms in each solid and ε\varepsilonε . What do we know about the system if we also know the total energy in each of the two objects?

Consider a system consisting of two Einstein solids PPP and QQQ in thermal contact. Assume that we know the number of atoms in each solid and ε\varepsilonε . What do we know about the system if we also know the total energy of the combined system?

What is the crucial characteristic of an Einstein solid that makes it easier to analyze in the context of this chapter than most other kinds of thermodynamic systems?

We can ignore an Einstein solid's zero-point energy because

Which of the following statements is true?

Suppose that we increase an NNN -atom Einstein solid's value of q=U/εq=U / \varepsilonq=U/ε from qqq to q+1q+1q+1 . By what factor does the value of its multiplicity Ω\OmegaΩ increase? (Choose whichever result is the closest, assuming that q>>1q>>1q>>1 and N>>1N>>1N>>1 .)

Suppose that you use a super-fast computer to measure a system's macropartition a billion times a second. What is the approximate probability of the least-probable macropartition that you might plausibly see in your lifetime? Select the closest response.

A hot object is placed in contact with a cold object. We observe that heat flows spontaneously from the hot object to the cold object, but not in the other direction. According to the argument in this chapter, this is so because

In a given macropartition, the total entropy of a system comprised of two or more parts is always equal to the sum of the entropies of those parts.

1062/1060=10010^{62} / 10^{60}=1001062/1060=100 . What is ln⁡1062/\ln 10^{62} /ln1062/ In 106010^{60}1060 ? (Select the closest response.)

Which system has the greater entropy, a puddle of water sitting on a table next to a chaotic scattering of salt crystals or the same amount of salt dissolved in an evenly distributed way in the puddle? Be prepared to describe your reasoning.

Object A's entropy increases quite a bit when we give it a certain tiny amount of energy, whereas object B's entropy increases by a much smaller amount with the same gift. Which has the higher temperature?

As a normal object's thermal energy UUU goes to zero, the slope of a graph of its entropy SSS versus UUU

Consider a system whose multiplicity is 1 , no matter how much energy one puts into it. What is such a system's temperature?

Consider a system consisting of two Einstein solids $P$ and $Q$ in thermal contact. Assume that we know the number of atoms in each solid and $\varepsilon$ . What do we know about the system if we also know the total energy in each of the two objects?

Consider a system consisting of two Einstein solids $P$ and $Q$ in thermal contact. Assume that we know the number of atoms in each solid and $\varepsilon$ . What do we know about the system if we also know the total energy of the combined system?

Suppose that we increase an $N$ -atom Einstein solid's value of $q=U / \varepsilon$ from $q$ to $q+1$ . By what factor does the value of its multiplicity $\Omega$ increase? (Choose whichever result is the closest, assuming that $q>>1$ and $N>>1$ .)

$10^{62} / 10^{60}=100$ . What is $\ln 10^{62} /$ In $10^{60}$ ? (Select the closest response.)

As a normal object's thermal energy $U$ goes to zero, the slope of a graph of its entropy $S$ versus $U$