Thermodynamics

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Thermodynamic Systems and Internal Energy

The internal energy of an ideal gas is the total kinetic energy of its particles, as ideal gas molecules have no intermolecular forces and therefore no potential energy.
For an ideal gas, the change in internal energy is directly proportional to the change in temperature: ΔU ∝ ΔT.
The change in internal energy can be calculated using ΔU = (3/2)NkBΔT, where N is the number of particles and kB is the Boltzmann constant.
Alternatively, ΔU = (3/2)nRΔT, where n is the number of moles and R is the molar gas constant.
When a gas is heated, its molecules move faster, increasing their kinetic energy and thus the internal energy.
In solids and liquids, molecules have both kinetic and potential energy due to intermolecular forces, but in an ideal gas, only kinetic energy contributes to internal energy.

Work Done by a Gas

When a gas expands, it does work on its surroundings; when it is compressed, work is done on the gas.
For a gas expanding at constant pressure, the work done is given by W = pΔV, where p is the pressure and ΔV is the change in volume.
The work done by a gas can be determined from the area under a pressure-volume (p-V) diagram.
On a p-V diagram, an expansion (volume increase) corresponds to positive work done by the gas, while a compression (volume decrease) corresponds to negative work done by the gas.
When the volume is constant, no work is done (W = 0).
The pressure p in the work done equation is the pressure of the surroundings, not the pressure of the gas itself.

Energy transfer and conservation

First Law of Thermodynamics

The First Law of Thermodynamics is a statement of conservation of energy: Q = ΔU + W, where Q is energy supplied by heating, ΔU is change in internal energy, and W is work done by the system.
Sign convention: Q is positive when heat is added to the system, W is positive when work is done by the system, and ΔU is positive when internal energy increases.
For a constant pressure (isobaric) process, work done is W = pΔV, so Q = ΔU + pΔV.
For a constant volume (isovolumetric) process, no work is done (W = 0), so Q = ΔU.
For an isothermal process, ΔU = 0, so Q = W.
For an adiabatic process, Q = 0, so W = −ΔU, meaning work done is at the expense of internal energy.

Entropy

Entropy is a measure of the number of possible arrangements of particles and their energies, i.e., a measure of disorder.
The order of entropy from most disordered to least is: gas > liquid > solid.
Entropy increases when a substance melts or boils, and decreases when it condenses or freezes.
Entropy increases when a solid dissolves in a solvent or when a gas diffuses, because particles become more spread out and there are more ways to arrange energy.
On a microscopic level, entropy is given by S = kB ln Ω, where Ω is the number of microstates.
The change in entropy when the number of microstates changes from Ω1 to Ω2 is ΔS = kB ln(Ω2/Ω1).

Calculating Changes in Entropy

At constant temperature, the change in entropy is ΔS = ΔQ/T, where ΔQ is heat added or removed and T is the temperature in Kelvin.
When heat is added (ΔQ > 0), entropy increases (ΔS > 0); when heat is removed (ΔQ < 0), entropy decreases (ΔS < 0).
For a reversible process that returns to the original state, ΔQ = 0 and ΔS = 0.
For a system of N distinguishable particles confined to one of two compartments, the initial number of microstates is Ω1 = 1, and after removing the partition, Ω2 = 2^N.
The change in entropy when a gas expands from volume V to 2V is ΔS = NkB ln 2.

Second Law of Thermodynamics

The Second Law of Thermodynamics states that the total entropy of an isolated system always increases; for a non-isolated system, the entropy of the Universe must increase.
Clausius form: Thermal energy cannot spontaneously transfer from a region of lower temperature to a region of higher temperature.
Kelvin form: When extracting energy from a heat reservoir, it is impossible to convert it all into work.
A reversible process is one where there is no overall change in entropy as the system and surroundings return to their original states.
An irreversible process results in an increase in entropy, and real isolated systems always undergo irreversible processes.
The entropy of a non-isolated system can decrease locally, but this is compensated by an equal or greater increase in the entropy of the surroundings.

Thermodynamic Processes

Isovolumetric (constant volume): W = 0, so Q = ΔU.
Isobaric (constant pressure): W = pΔV, so Q = ΔU + pΔV.
Isothermal (constant temperature): ΔU = 0, so Q = W.
Adiabatic (no heat transfer): Q = 0, so W = −ΔU.
For a monatomic ideal gas undergoing an adiabatic process, pV^(5/3) = constant, which can be used to find changes in pressure, volume, and temperature.
In an adiabatic expansion, the gas does work and cools down; in an adiabatic compression, work is done on the gas and it heats up.

Heat Engines

A heat engine converts thermal energy into mechanical work through a cyclic process.
The cycle involves: extracting heat QH from a hot reservoir, doing work W, and releasing heat QC to a cold reservoir.
The net work done by the engine is W = QH − QC.
The efficiency of a heat engine is η = W/QH = 1 − QC/QH.
On a p-V diagram, a cyclic process forms a closed loop, and the area inside the loop equals the net work done per cycle.

The Carnot Cycle

The Carnot cycle is an idealised reversible cycle with the maximum possible efficiency.
It consists of four stages: isothermal expansion, adiabatic expansion, isothermal compression, and adiabatic compression.
During isothermal expansion, the gas absorbs heat QH at temperature TH; during isothermal compression, it releases heat QC at temperature TC.
During adiabatic stages, no heat is transferred, but temperature changes between TH and TC.
The maximum theoretical efficiency of a Carnot engine is ηC = 1 − TC/TH, where temperatures are in Kelvin.
For a Carnot cycle, QH/TH = QC/TC, which leads to the efficiency formula.
Any heat engine operating between two temperatures cannot exceed the Carnot efficiency; otherwise, it violates the second law of thermodynamics.

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练习题

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1.What is internal energy?
Easy
- AThe total kinetic and potential energy of the particles in a system
- BThe total energy transferred to a system by heating
- CThe energy transferred when work is done on a system
- DThe average kinetic energy of the particles in a system
2.Which of the following is the correct definition of entropy?
Easy
- AA measure of the disorder of a system
- BA measure of the average kinetic energy of particles
- CA measure of the total energy of a system
- DA measure of the work done by a system
3.In an adiabatic process, no heat is transferred into or out of the system.
Easy
True or false?
4.The second law of thermodynamics states that the total entropy of an isolated system always decreases.
Easy
True or false?
5.In which process is a gas doing the least amount of work?
Medium
- AIsothermal expansion
- BIsobaric expansion
- CIsovolumetric heating
- DAdiabatic expansion
6.A gas is kept in a closed container at room temperature. A lit Bunsen burner is then placed under this container. Which row correctly describes the changes the gas undergoes?
Medium
- AWork done on gas: positive; Change in internal energy: positive
- BWork done on gas: zero; Change in internal energy: zero
- CWork done on gas: positive; Change in internal energy: zero
- DWork done on gas: zero; Change in internal energy: positive
7.A hypothetical car is running on a Carnot engine. Once the car has been driving for a while, the gas in the engine has a minimum temperature of 275 K. The engine has an efficiency of 0.75. What is the temperature range of the gas in the engine?
Medium
- A92 K
- B367 K
- C825 K
- D1100 K
8.An ideal gas has an initial pressure of 150 kPa, an initial temperature of 27°C, and an initial volume of 0.2 m³. During an isobaric compression, the volume of the gas decreases by 0.005 m³. What is the final pressure of the gas?
Medium
- A50 kPa
- B150 kPa
- C240 kPa
- D480 kPa

更多题目

9.A refrigerator in its surroundings does work and freezes a small amount of liquid water. Which row of the table is correct?
Medium
- AEntropy of water: increases; Temperature of surroundings: increases; Entropy of surroundings: increases
- BEntropy of water: decreases; Temperature of surroundings: increases; Entropy of surroundings: increases
- CEntropy of water: stays the same; Temperature of surroundings: decreases; Entropy of surroundings: decreases
- DEntropy of water: decreases; Temperature of surroundings: increases; Entropy of surroundings: stays the same
10.A simple heat engine contains 3 moles of an ideal gas. When the gas is cooled, the mean kinetic energy per particle drops by 3.11 × 10⁻²¹ J. Additionally, 2 kJ of work is done on the gas. What is the thermal energy transferred from the gas to its surroundings?
Medium
- A2000 J
- B3620 J
- C5620 J
- D7620 J
11.An electrical heater has a current of 1.5 A passing through it and a potential difference of 200 V across it. It heats a gas in a piston at a constant pressure of 30 kPa. After 15 s the volume of the gas has increased by 0.020 m³ and its internal energy has increased by 2000 J. The electrical heater is 70% efficient. What proportion of the thermal energy output by the electric heater is transferred to the gas?
Hard
- A0.44
- B0.64
- C0.70
- D0.83
12.At 78.0°C, 12.0 g of liquid ammonia is converted to a gaseous state. The entropy change of this process is 0.0470 J K⁻¹. What is the latent heat of vaporisation of ammonia?
Hard
- A16.5 J kg⁻¹
- B39.2 J kg⁻¹
- C306 J kg⁻¹
- D1370 J kg⁻¹
13.In an adiabatic process, a gas in a piston does work. The cylindrical chamber increases in length from 1.5 m to 290 cm. Which expression, in terms of original pressure p, most closely represents the new pressure?
Hard
- Ap/5
- Bp/3
- C3p
- D5p
14.An idealised heat engine has an efficiency of η and is supplied with heat energy QH from an external hot reservoir. The cycle is described as follows: Stage 1: The gas undergoes isobaric compression at a pressure of p1. The volume decreases by z and the final temperature is T1. Stage 2: The gas then undergoes isovolumetric heating. The final temperature is T2. Stage 3: The gas undergoes isobaric expansion. The volume increases by z. Stage 4: Finally, the gas undergoes isovolumetric cooling and returns to its state at the beginning of stage 1. Which is the correct expression for z?
Hard
- Aη QH T1 / (p1 (T2 - T1))
- Bη QH / (p1 (T2 - T1))
- Cη QH T1 / (p1 (T1 - T2))
- Dη QH / (p1 (T1 - T2))
15.An ideal monatomic gas undergoes the process pV^(5/3) = k, where k is a constant. Which of the following statements are true? (select all that apply)
Medium
- ANo thermal energy is transferred from the gas to its surroundings
- BThe internal energy of the gas is constant
- CWork is done on the gas
- DThe temperature of the gas changes
- EThe entropy of the gas changes
16.Match each thermodynamic process to the quantity that remains constant during it.
Medium
- Isovolumetric
- Isobaric
- Isothermal
- Adiabatic
- Volume
- Pressure
- Temperature
- Thermal energy transferred (Q = 0)
17.A gas in a cylinder is compressed at constant pressure. Which row correctly describes the work done and the change in internal energy if no heat is exchanged?
Medium
- AWork is done on the gas; internal energy increases
- BWork is done by the gas; internal energy decreases
- CWork is done on the gas; internal energy decreases
- DWork is done by the gas; internal energy increases
18.Which of the following processes result in an increase in the entropy of the system? (select all that apply)
Medium
- AA solid melting
- BA gas condensing into a liquid
- CA gas expanding isothermally
- DA liquid freezing
- EA solid dissolving in a solvent
19.Place the following states of matter in order of increasing entropy, from lowest to highest.
Medium
- Solid
- Liquid
- Gas
20.A Carnot cycle consists of four stages. Which of the following statements about the cycle are correct? (select all that apply)
Hard
- AThe isothermal expansion involves heat QH entering the gas from a hot reservoir
- BThe adiabatic expansion involves no heat transfer but the temperature falls from TH to TC
- CThe isothermal compression involves heat QC leaving the gas to a cold reservoir
- DThe net work done in one cycle is equal to the area enclosed by the loop on a p-V diagram
- EThe efficiency of the Carnot cycle depends on the working substance used
21.An ideal gas expands isothermally from volume V to 2V. The change in entropy is given by which expression?
Hard
- AΔS = N kB ln 2
- BΔS = N kB ln (1/2)
- CΔS = 0
- DΔS = N kB
22.A heat engine operates between a hot reservoir at TH and a cold reservoir at TC. Which of the following statements are true for a Carnot engine? (select all that apply)
Hard
- AThe efficiency is given by η = 1 - TC/TH
- BThe efficiency can be greater than 1 if TH is very large
- CThe efficiency increases as TH increases
- DThe efficiency is independent of the working substance
- EThe efficiency is 100% if TC = 0 K
23.Which of the following statements about the second law of thermodynamics are correct? (select all that apply)
Medium
- AThe total entropy of an isolated system always increases
- BHeat can spontaneously flow from a colder to a hotter body
- CIt is impossible to convert all heat from a reservoir into work
- DThe entropy of a non-isolated system can decrease locally
- EThe entropy of the universe must increase
24.Which of the following are true about the first law of thermodynamics? (select all that apply)
Medium
- AIt is based on the principle of conservation of energy
- BIt applies only to gases
- CIt can be written as Q = ΔU + W
- DWork done by the system is positive
- EIt states that entropy always increases
25.A gas undergoes a cyclic process. Which of the following are true? (select all that apply)
Hard
- AThe net change in internal energy over one complete cycle is zero
- BThe net work done equals the area enclosed by the cycle on a p-V diagram
- CThe net heat transfer over the cycle is zero
- DThe entropy of the gas returns to its initial value after one cycle
- EThe temperature of the gas is the same at the start and end of the cycle
26.A fixed mass of ideal gas is heated at constant volume. Which row correctly describes the work done and the change in internal energy?
Medium
- AWork done is zero; internal energy increases
- BWork done is positive; internal energy increases
- CWork done is zero; internal energy is unchanged
- DWork done is negative; internal energy increases
27.An ideal gas expands isothermally from volume V to 2V. Which expression gives the change in entropy?
Medium
- ANkB ln 2
- BNkB ln (1/2)
- CNkB / 2
- D2 NkB
28.A Carnot engine operates between a hot reservoir at 600 K and a cold reservoir at 300 K. It absorbs 1200 J from the hot reservoir each cycle. How much heat is rejected to the cold reservoir per cycle?
Hard
- A600 J
- B300 J
- C1200 J
- D900 J
29.An ideal monatomic gas is compressed adiabatically. Which statement is correct?
Medium
- AThe temperature of the gas increases
- BThe temperature of the gas decreases
- CThe internal energy of the gas is unchanged
- DHeat flows out of the gas to the surroundings
30.A gas in a cylinder is heated at constant pressure. Which of the following statements are correct? (select all that apply)
Medium
- AWork is done by the gas as it expands
- BThe internal energy of the gas increases
- CThe temperature of the gas increases
- DThe change in volume is zero
- ENo heat is transferred to the gas
31.A sealed container of gas is divided by a partition. The partition is removed and the gas fills the whole container. Which of the following statements are correct? (select all that apply)
Hard
- AThe number of possible microstates of the gas increases
- BThe entropy of the gas increases
- CThe process is irreversible
- DThe entropy of the gas decreases
- EThe temperature of the gas must increase
32.In an adiabatic process, the entropy of an ideal gas remains constant.
Easy
True or false?
33.During an isothermal expansion of an ideal gas, the internal energy of the gas increases.
Easy
True or false?
34.The work done by a gas during a thermodynamic process can be found from the area under the line on a pressure-volume diagram.
Medium
True or false?
35.Place the four stages of a Carnot cycle in the correct order, starting from the isothermal expansion.
Medium
- Isothermal expansion
- Adiabatic expansion
- Isothermal compression
- Adiabatic compression
36.If the hot reservoir is at 600 K, what is the minimum possible temperature of the cold reservoir for this engine to be thermodynamically possible?
Hard
- A375 K
- B225 K
- C500 K
- D300 K
37.A student wants to measure the latent heat of vaporisation of a liquid using the equation ΔS = ΔQ/T. Which quantity must be measured directly and at what temperature should it be recorded?
Easy
- AThe heat supplied and the constant boiling temperature of the liquid
- BThe heat supplied and the room temperature
- CThe pressure and the constant boiling temperature
- DThe volume change and the room temperature
38.A sealed rigid container holds an ideal gas. A thermometer shows the temperature rising steadily while a pressure gauge shows the pressure rising in proportion. Which of the following describes this process and the work done?
Medium
- AIsovolumetric; no work is done
- BIsobaric; work is done by the gas
- CIsothermal; work is done on the gas
- DAdiabatic; work is done on the gas
39.An ideal monatomic gas at 300 K is compressed adiabatically to half its volume. Which expression gives the new temperature?
Medium
- A300 × 2^(2/3) K
- B300 × 2^(5/3) K
- C300 × (1/2)^(2/3) K
- D300 × (1/2)^(5/3) K
40.A gas in a cylinder is compressed very rapidly so that no heat is exchanged with the surroundings. Which of the following statements are correct? (select all that apply)
Medium
- AThe process is adiabatic
- BThe temperature of the gas increases
- CThe internal energy of the gas increases
- DThe entropy of the gas increases
- EWork is done by the gas
41.Complete the sentence about the first law of thermodynamics.
Easy
The first law of thermodynamics is based on the principle of conservation of ____.
42.Name the four main thermodynamic processes.
Easy
The four main thermodynamic processes are ____, ____, ____, and ____.
43.Complete the statement of the second law of thermodynamics.
Medium
In every process, the total entropy of an isolated system always ____.
44.An ideal gas expands at a constant pressure of 1.0 × 10⁵ Pa. Its volume increases from 0.015 m³ to 0.030 m³. Calculate the work done by the gas.
Medium
45.A monatomic gas contains 5.2 × 10²⁵ atoms and its temperature increases from 260 K to 300 K. Calculate the change in internal energy. Use kB = 1.38 × 10^-23 J K^-1.
Hard
46.An ideal monatomic gas expands adiabatically from a pressure of 7.5 × 10⁵ Pa and volume 1.8 × 10^-3 m³ to a volume of 4.2 × 10^-3 m³. Calculate the new pressure. Give your answer in Pa to 2 significant figures.
Hard
47.An ideal gas expands at a constant pressure of 1.5 × 10⁵ Pa. Its volume increases from 0.020 m³ to 0.050 m³. Calculate the work done by the gas.
Medium
48.An ideal monatomic gas expands adiabatically from a pressure of 2.0 × 10⁵ Pa and volume 1.0 × 10^-3 m³ to a volume of 8.0 × 10^-3 m³. Calculate the new pressure of the gas.
Hard
49.A Carnot engine operates between a hot reservoir at 500 K and a cold reservoir at 350 K. Calculate its maximum theoretical efficiency as a fraction (give your answer as a decimal).
Hard
50.An ideal gas is heated at constant volume. The number of particles is 3.0 × 10²³ and the temperature rises by 40 K. Calculate the change in internal energy. Use kB = 1.38 × 10^-23 J K^-1.
Hard
51.Complete the equation for the maximum theoretical efficiency of a heat engine.
Medium
The maximum theoretical efficiency of a heat engine is given by ηC = 1 − ____ / ____
52.What name is given to a thermodynamic process in which no heat is transferred into or out of the system?
Medium
53.Explain why the work done on the gas during the process CD is less than the work done by the gas during the process AB in a Carnot cycle.
Hard
54.State and explain whether the balloon increases or decreases in size when a monatomic gas in a balloon increases in temperature from 260 K to 300 K and 25 kJ of thermal energy is transferred to the gas.
Hard
55.A freezer door is opened while the freezer is switched on and placed in a sealed room. State and explain what happens to the entropy of the room.
Hard
56.A gas in a cylinder is compressed quickly so that no heat is exchanged with the surroundings. State and explain what happens to the temperature of the gas.
Hard
57.An engineer claims a heat engine operating between 500 K and 300 K can have an efficiency of 75%. Discuss whether this is possible, referring to the second law of thermodynamics.
Hard
58.An ideal gas expands at a constant pressure of 1.0 × 10⁵ Pa. Its volume increases from 0.015 m³ to 0.030 m³. Calculate the work done by the gas.
Medium
- A1800 J
- B1350 J
- C1650 J
- D1500 J
59.A monatomic gas contains 5.2 × 10²⁵ atoms and its temperature increases from 260 K to 300 K. Calculate the change in internal energy. Use kB = 1.38 × 10^-23 J K^-1.
Hard
- A14352 J
- B12917 J
- C15787 J
- D17222 J
60.An ideal monatomic gas expands adiabatically from a pressure of 7.5 × 10⁵ Pa and volume 1.8 × 10^-3 m³ to a volume of 4.2 × 10^-3 m³. Calculate the new pressure. Give your answer in Pa to 2 significant figures.
Hard
- A180000 Pa
- B198000 Pa
- C162000 Pa
- D216000 Pa
61.An ideal gas expands at a constant pressure of 1.5 × 10⁵ Pa. Its volume increases from 0.020 m³ to 0.050 m³. Calculate the work done by the gas.
Medium
- A4500 J
- B4950 J
- C5400 J
- D4050 J
62.An ideal monatomic gas expands adiabatically from a pressure of 2.0 × 10⁵ Pa and volume 1.0 × 10^-3 m³ to a volume of 8.0 × 10^-3 m³. Calculate the new pressure of the gas.
Hard
- A37500 Pa
- B31250 Pa
- C34375 Pa
- D28125 Pa
63.A Carnot engine operates between a hot reservoir at 500 K and a cold reservoir at 350 K. Calculate its maximum theoretical efficiency as a fraction (give your answer as a decimal).
Hard
- A1.3
- B2.3
- C-0.7
- D0.3
64.An ideal gas is heated at constant volume. The number of particles is 3.0 × 10²³ and the temperature rises by 40 K. Calculate the change in internal energy. Use kB = 1.38 × 10^-23 J K^-1.
Hard
- A223.6 J
- B273.2 J
- C298 J
- D248.4 J

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