Question 1

A container holds 1 million helium molecules at $300 \mathrm{~K}$. If we double the gas's temperature to $600 \mathrm{~K}$, by about what factor does its multiplicity increase?

Accepted Answer

The multiplicity of an ideal gas is proportional to $T^{3N/2}$, where $N$ is the number of molecules. Doubling the temperature from 300 K to 600 K increases the multiplicity by a factor of $2^{3N/2}$. With $N = 1,000,000$, the factor is $2^{1,500,000}$.

Question 2

How does the entropy of $1 \mathrm{~mol}$ of helium gas in a container at room temperature compare to $1 \mathrm{~mol}$ of argon gas in an identical container at the same temperature?

Accepted Answer

The answer of How does the entropy of $1 \mathrm{~mol}$...

Question 3

Suppose that we allow an ideal gas containing $N$ molecules to expand slowly and adiabatically to twice its initial volume. By what factor does its multiplicity increase?

Accepted Answer

In an adiabatic process, there is no heat exchange with the surroundings, and for an ideal gas, the entropy remains constant. Since multiplicity is related to entropy, it does not change.

Question 4

Suppose that we allow an ideal gas containing $N$ atoms to expand slowly and isothermally to twice its initial volume. By what factor does its multiplicity increase?

Accepted Answer

The multiplicity of an ideal gas, which is a measure of the number of microstates corresponding to a given macrostate, increases exponentially with volume for a given number of particles and temperature. When an ideal gas expands isothermally to twice its initial volume, its multiplicity increases by a factor of $2^N$, where $N$ is the number of particles in the gas. This is because each particle has twice as many positions it can occupy, leading to an exponential increase in the total number of microstates.

Question 5

Suppose we hold a sample of helium gas at a constant temperature while we slowly compress it in a cylinder until its volume has decreased by a factor of 4 . During this process, the gas's entropy

Accepted Answer

Entropy decreases when a gas is compressed at constant temperature because the gas molecules have less volume to occupy, leading to a decrease in the number of possible microstates and thus a decrease in entropy.

Question 6

Consider each of the following processes. Is the process consistent (C) or inconsistent (D) with the limitations on equation T8.12 $d S=\frac{d Q}{T}$ (that $N$ be fixed and that the process not involve any energy transfer that is neither heat nor work due to a quasistatic volume change)?
-a) A glass of milk is spilled on the floor.

Accepted Answer

The process of spilling milk involves a change in the number of particles in the system (as the milk spreads out, some of it may evaporate or be absorbed by the floor), and it does not involve a quasistatic volume change or a clear definition of system temperature, making it inconsistent with the limitations of equation $d S=\frac{d Q}{T}$.

Question 7

Consider each of the following processes. Is the process consistent (C) or inconsistent (D) with the limitations on equation T8.12 $d S=\frac{d Q}{T}$ (that $N$ be fixed and that the process not involve any energy transfer that is neither heat nor work due to a quasistatic volume change)?
-b) A closed vial of gas is heated with a flame.

Accepted Answer

The process of heating a closed vial of gas with a flame typically involves transferring heat to the gas without changing the number of particles in the system (N is fixed) and without involving energy transfer that is neither heat nor work due to a quasistatic volume change. Therefore, it is consistent with the limitations on equation $d S=\frac{d Q}{T}$.

Question 8

Consider each of the following processes. Is the process consistent (C) or inconsistent (D) with the limitations on equation T8.12 $d S=\frac{d Q}{T}$ (that $N$ be fixed and that the process not involve any energy transfer that is neither heat nor work due to a quasistatic volume change)?
-c) A pan of water boils on the stove.

Accepted Answer

The process of boiling water involves a change in the number of moles of water as it transitions from liquid to gas, violating the condition that $N$ (the number of particles) be fixed.

Question 9

Consider each of the following processes. Is the process consistent (C) or inconsistent (D) with the limitations on equation T8.12 $d S=\frac{d Q}{T}$ (that $N$ be fixed and that the process not involve any energy transfer that is neither heat nor work due to a quasistatic volume change)?
-Consider two $q$-vectors $|u\rangle$ and $|w\rangle$ with possibly complex components. Their inner product $\langle u \mid w\rangle$ is a real number.

Accepted Answer

The answer of Consider each of the following processes. Is...

Question 10

Consider each of the following processes. Is the process consistent (C) or inconsistent (D) with the limitations on equation T8.12 $d S=\frac{d Q}{T}$ (that $N$ be fixed and that the process not involve any energy transfer that is neither heat nor work due to a quasistatic volume change)?
-

Accepted Answer

The answer of Consider each of the following processes. Is...

Question 11

Suppose that a gas expands in a cylinder, pushing back a piston with a speed of $20 \mathrm{~m} / \mathrm{s}$ (about $45 \mathrm{mi} / \mathrm{h}$ ). We can reasonably consider such an expansion to be quasistatic.

Accepted Answer

The expansion is quasistatic, which means that it occurs slowly enough that the gas is always in equilibrium. In equilibrium, the pressure and volume of the gas are constant, so the piston moves at a constant speed.

Question 12

In a tub, $100 \mathrm{~kg}$ of water at $30^{\circ} \mathrm{C}$ absorbs $100 \mathrm{~J}$ from an electric heater. We can reasonably consider the water to be a reservoir in this process.

Accepted Answer

The amount of heat absorbed by the water (100 J) is relatively small compared to the large mass of water (100 kg), and thus the temperature change of the water would be negligible, allowing it to act as a thermal reservoir.

Question 13

We slowly heat $100 \mathrm{~g}$ of water from 5 to $25^{\circ} \mathrm{C}$. What is its change in entropy? (For water, $c=4186 \mathrm{~kg}$.)

Accepted Answer

The answer of We slowly heat $100 \mathrm{~g}$ of water...

Question 14

We bring a hot object into contact with a cold object. The hot object's entropy decreases by $1.5 \mathrm{~J} / \mathrm{K}$ as it comes into equilibrium with the cold object. By how much does the cold object's entropy increase?

Accepted Answer

The entropy increase of the cold object is more than the decrease in entropy of the hot object due to the second law of thermodynamics, which states that the total entropy of an isolated system can never decrease over time.

Question 15

We drop a stone of mass $m$ from a height $h$ into a bucket of water. What would be a suitable replacement process for this non-quasistatic process?

Accepted Answer

The answer of We drop a stone of mass $m$...

Question 16

A sample of helium gas with an initial temperature of $T_{i}$ in an insulated cylinder with initial volume $V_{i}$ is suddenly and violently compressed to one-half its volume. The absolute temperature of the gas after the compression is observed to have doubled. (Note: In a quasistatic adiabatic compression of a monatomic gas, $\mathrm{TV}^{2 / 3}=$ constant.) A suitable replacement process for this process would be

Accepted Answer

For an adiabatic compression of a monatomic gas, $TV^{2/3}$ is constant. Therefore, if the volume is halved, the temperature should increase by a factor of 8. Since the temperature only doubles, the compression must not be adiabatic. The only choice that does not involve an adiabatic compression is D.

Question 17

Classify the following hypothetical perpetual motion machines as being perpetual motion machines of the first kind $(A)$ or the second kind (B).
-(a) An electric car runs off a battery which drives the front wheels. The car's rear wheels drive a generator which recharges the battery.

Accepted Answer

A perpetual motion machine of the first kind violates the first law of thermodynamics by creating energy from nothing, as in the case of the electric car that supposedly recharges its own battery indefinitely without any external energy source.

Question 18

Classify the following hypothetical perpetual motion machines as being perpetual motion machines of the first kind $(A)$ or the second kind (B).
-(b) An electric car runs off a battery. When the driver wants to slow down the car, instead of applying the brakes he or she throws a switch that connects the wheels to a generator that converts the car's kinetic energy back to energy in the battery.

Accepted Answer

The answer of Classify the following hypothetical perpetual motion machines...

Question 19

Classify the following hypothetical perpetual motion machines as being perpetual motion machines of the first kind $(A)$ or the second kind (B).
-(c) An engine's tank is filled with water. When the engine operates, it slowly freezes the water in the tank, converting the energy released to mechanical energy.

Accepted Answer

This hypothetical engine violates the second law of thermodynamics by converting thermal energy entirely into mechanical work without any loss, classifying it as a perpetual motion machine of the second kind.

Question 20

Classify the following hypothetical perpetual motion machines as being perpetual motion machines of the first kind $(A)$ or the second kind (B).
-(d) Compressed air from a tank blows on a windmill. The windmill is connected to a generator that produces electrical energy. Part of that electrical energy is used to compress more air into the tank.

Accepted Answer

The answer of Classify the following hypothetical perpetual motion machines...

Quiz 6: Some Processes Are Irreversible Thermal Physics

A container holds 1 million helium molecules at $300 \mathrm{~K}$ . If we double the gas's temperature to $600 \mathrm{~K}$ , by about what factor does its multiplicity increase?

How does the entropy of $1 \mathrm{~mol}$ of helium gas in a container at room temperature compare to $1 \mathrm{~mol}$ of argon gas in an identical container at the same temperature?

Suppose that we allow an ideal gas containing $N$ molecules to expand slowly and adiabatically to twice its initial volume. By what factor does its multiplicity increase?

Suppose that we allow an ideal gas containing $N$ atoms to expand slowly and isothermally to twice its initial volume. By what factor does its multiplicity increase?

Suppose we hold a sample of helium gas at a constant temperature while we slowly compress it in a cylinder until its volume has decreased by a factor of 4 . During this process, the gas's entropy

Suppose that a gas expands in a cylinder, pushing back a piston with a speed of $20 \mathrm{~m} / \mathrm{s}$ (about $45 \mathrm{mi} / \mathrm{h}$ ). We can reasonably consider such an expansion to be quasistatic.

In a tub, $100 \mathrm{~kg}$ of water at $30^{\circ} \mathrm{C}$ absorbs $100 \mathrm{~J}$ from an electric heater. We can reasonably consider the water to be a reservoir in this process.

We slowly heat $100 \mathrm{~g}$ of water from 5 to $25^{\circ} \mathrm{C}$ . What is its change in entropy? (For water, $c=4186 \mathrm{~kg}$ .)

We bring a hot object into contact with a cold object. The hot object's entropy decreases by $1.5 \mathrm{~J} / \mathrm{K}$ as it comes into equilibrium with the cold object. By how much does the cold object's entropy increase?

We drop a stone of mass $m$ from a height $h$ into a bucket of water. What would be a suitable replacement process for this non-quasistatic process?

Classify the following hypothetical perpetual motion machines as being perpetual motion machines of the first kind $(A)$ or the second kind (B) . -(a) An electric car runs off a battery which drives the front wheels. The car's rear wheels drive a generator which recharges the battery.

Quiz 6: Some Processes Are Irreversible Thermal Physics

A container holds 1 million helium molecules at 300 K300 \mathrm{~K}300 K . If we double the gas's temperature to 600 K600 \mathrm{~K}600 K , by about what factor does its multiplicity increase?

How does the entropy of 1 mol1 \mathrm{~mol}1 mol of helium gas in a container at room temperature compare to 1 mol1 \mathrm{~mol}1 mol of argon gas in an identical container at the same temperature?

Suppose that we allow an ideal gas containing NNN molecules to expand slowly and adiabatically to twice its initial volume. By what factor does its multiplicity increase?

Suppose that we allow an ideal gas containing NNN atoms to expand slowly and isothermally to twice its initial volume. By what factor does its multiplicity increase?

Suppose we hold a sample of helium gas at a constant temperature while we slowly compress it in a cylinder until its volume has decreased by a factor of 4 . During this process, the gas's entropy

Suppose that a gas expands in a cylinder, pushing back a piston with a speed of 20 m/s20 \mathrm{~m} / \mathrm{s}20 m/s (about 45mi/h45 \mathrm{mi} / \mathrm{h}45mi/h ). We can reasonably consider such an expansion to be quasistatic.

In a tub, 100 kg100 \mathrm{~kg}100 kg of water at 30∘C30^{\circ} \mathrm{C}30∘C absorbs 100 J100 \mathrm{~J}100 J from an electric heater. We can reasonably consider the water to be a reservoir in this process.

We slowly heat 100 g100 \mathrm{~g}100 g of water from 5 to 25∘C25^{\circ} \mathrm{C}25∘C . What is its change in entropy? (For water, c=4186 kgc=4186 \mathrm{~kg}c=4186 kg .)

We bring a hot object into contact with a cold object. The hot object's entropy decreases by 1.5 J/K1.5 \mathrm{~J} / \mathrm{K}1.5 J/K as it comes into equilibrium with the cold object. By how much does the cold object's entropy increase?

We drop a stone of mass mmm from a height hhh into a bucket of water. What would be a suitable replacement process for this non-quasistatic process?

Classify the following hypothetical perpetual motion machines as being perpetual motion machines of the first kind (A) (A) (A) or the second kind (B) . -(a) An electric car runs off a battery which drives the front wheels. The car's rear wheels drive a generator which recharges the battery.

A container holds 1 million helium molecules at $300 \mathrm{~K}$ . If we double the gas's temperature to $600 \mathrm{~K}$ , by about what factor does its multiplicity increase?

How does the entropy of $1 \mathrm{~mol}$ of helium gas in a container at room temperature compare to $1 \mathrm{~mol}$ of argon gas in an identical container at the same temperature?

Suppose that we allow an ideal gas containing $N$ molecules to expand slowly and adiabatically to twice its initial volume. By what factor does its multiplicity increase?

Suppose that we allow an ideal gas containing $N$ atoms to expand slowly and isothermally to twice its initial volume. By what factor does its multiplicity increase?

Suppose that a gas expands in a cylinder, pushing back a piston with a speed of $20 \mathrm{~m} / \mathrm{s}$ (about $45 \mathrm{mi} / \mathrm{h}$ ). We can reasonably consider such an expansion to be quasistatic.

In a tub, $100 \mathrm{~kg}$ of water at $30^{\circ} \mathrm{C}$ absorbs $100 \mathrm{~J}$ from an electric heater. We can reasonably consider the water to be a reservoir in this process.

We slowly heat $100 \mathrm{~g}$ of water from 5 to $25^{\circ} \mathrm{C}$ . What is its change in entropy? (For water, $c=4186 \mathrm{~kg}$ .)

We bring a hot object into contact with a cold object. The hot object's entropy decreases by $1.5 \mathrm{~J} / \mathrm{K}$ as it comes into equilibrium with the cold object. By how much does the cold object's entropy increase?

We drop a stone of mass $m$ from a height $h$ into a bucket of water. What would be a suitable replacement process for this non-quasistatic process?

Classify the following hypothetical perpetual motion machines as being perpetual motion machines of the first kind $(A)$ or the second kind (B) . -(a) An electric car runs off a battery which drives the front wheels. The car's rear wheels drive a generator which recharges the battery.