IB Physics HL | Unit B4: Thermodynamics

Show all working in the space below each question. Marks are awarded for correct method as well as final answers. State the IB sign convention before applying the first law: $Q$ is heat added to the gas; $W$ is work done by the gas. Keep all temperatures in kelvin.在每题下方空白处写出全部解题过程。方法分（method marks）与最终答案同等重要。套用第一定律前先写明 IB 符号约定：$Q$ 为加给气体的热量；$W$ 为气体对外做的功。所有温度一律用开尔文。

Q1MEDIUM Paper 1 HL ONLY first law, sign convention第一定律与符号约定 [4 marks]

During one process a gas is given $360\ \mathrm{J}$ of heat while it is compressed, the surroundings doing $150\ \mathrm{J}$ of work on the gas. Use the IB form $Q = \Delta U + W$, where $W$ is the work done by the gas.在某过程中，气体被压缩，外界对其做 $150\ \mathrm{J}$ 的功，同时气体获得 $360\ \mathrm{J}$ 热量。用 IB 形式 $Q = \Delta U + W$，其中 $W$ 为气体对外做的功。

(a) State, with the correct sign, the value of $Q$ and of $W$ for this process.写出该过程中 $Q$ 与 $W$ 的取值及正确符号。 [2]

(b) Hence calculate the change in internal energy $\Delta U$ of the gas, and state whether the gas warms or cools.由此计算气体内能的变化 $\Delta U$，并说明气体升温还是降温。 [2]

Q2MEDIUM Paper 1 HL ONLY isobaric work + first law等压做功与第一定律 [4 marks]

A gas at a constant pressure of $1.8\times 10^{5}\ \mathrm{Pa}$ expands from $2.0\times 10^{-3}\ \mathrm{m^{3}}$ to $5.0\times 10^{-3}\ \mathrm{m^{3}}$. During the expansion $900\ \mathrm{J}$ of heat is supplied to the gas.气体在恒压 $1.8\times 10^{5}\ \mathrm{Pa}$ 下由 $2.0\times 10^{-3}\ \mathrm{m^{3}}$ 膨胀到 $5.0\times 10^{-3}\ \mathrm{m^{3}}$。膨胀过程中向气体供给 $900\ \mathrm{J}$ 热量。

(a) Calculate the work done by the gas.计算气体所做的功。 [2]

(b) Hence determine the change in internal energy of the gas.由此求气体内能的变化。 [2]

Q3HARD Paper 1 HL ONLY four named processes四种命名过程 [6 marks]

For each scenario below, identify the type of process (isothermal, isobaric, isochoric or adiabatic), reduce the first law $Q = \Delta U + W$ to the form it takes for that process, and find the requested quantity.对下面每种情形，判断过程类型（等温、等压、等容或绝热），把第一定律 $Q = \Delta U + W$ 化为该过程下的形式，并求所要的量。

(a) An ideal gas expands at constant temperature and absorbs $300\ \mathrm{J}$ of heat. Find the work done by the gas.理想气体在恒温下膨胀，吸收 $300\ \mathrm{J}$ 热量。求气体所做的功。 [2]

(b) A gas in a rigid sealed container is heated, gaining $300\ \mathrm{J}$ of thermal energy. Find the change in its internal energy.刚性密封容器中的气体被加热，获得 $300\ \mathrm{J}$ 热能。求其内能变化。 [2]

(c) A gas is compressed suddenly inside a well-insulated cylinder, the surroundings doing $250\ \mathrm{J}$ of work on it with no heat exchange. Find the change in its internal energy and state whether it warms or cools.气体在隔热良好的气缸内被突然压缩，外界对其做 $250\ \mathrm{J}$ 功且无热量交换。求其内能变化，并说明升温还是降温。 [2]

Q4HARD Paper 1 HL ONLY entropy of heat flow + second law热流熵变与第二定律 [6 marks]

$6000\ \mathrm{J}$ of heat flows from a large hot reservoir at $600\ \mathrm{K}$ to a large cold reservoir at $300\ \mathrm{K}$. Both reservoirs are so large that their temperatures do not change. Use $\Delta S = \Delta Q / T$.$6000\ \mathrm{J}$ 热量从 $600\ \mathrm{K}$ 的大热源流向 $300\ \mathrm{K}$ 的大冷源。两热源都足够大，温度不变。用 $\Delta S = \Delta Q / T$。

(a) Calculate the entropy change of the hot reservoir and of the cold reservoir.分别计算热源与冷源的熵变。 [3]

(b) Calculate the total entropy change and state, with reference to the second law, whether this heat flow can occur spontaneously.计算总熵变，并结合第二定律说明该热流能否自发发生。 [2]

(c) State what the total entropy change would be if the same $6000\ \mathrm{J}$ instead flowed from the cold reservoir to the hot one, and what this implies.若同样的 $6000\ \mathrm{J}$ 改为从冷源流向热源，写出总熵变会是多少，以及这说明了什么。 [1]

Q5HARD Paper 1 HL ONLY heat-engine efficiency (two forms)热机效率（两种形式） [4 marks]

In each cycle a heat engine takes in $1500\ \mathrm{J}$ from its hot reservoir and rejects $900\ \mathrm{J}$ to its cold reservoir.每个循环中，一台热机从热源吸收 $1500\ \mathrm{J}$，向冷源排出 $900\ \mathrm{J}$。

(a) Calculate the useful work output per cycle, explaining why $\Delta U = 0$ over a complete cycle.计算每循环的有用输出功，并解释为何一个完整循环 $\Delta U = 0$。 [2]

(b) Calculate the thermal efficiency, and verify your answer using the rejected-heat form $\eta = 1 - Q_{\mathrm{out}} / Q_{\mathrm{in}}$.计算热效率，并用排热式 $\eta = 1 - Q_{\mathrm{out}} / Q_{\mathrm{in}}$ 验证答案。 [2]

Q6MEDIUM Paper 1 HL ONLY Carnot limit + plausibility check卡诺极限与可行性判断 [6 marks]

A heat engine operates between a hot reservoir at $500\ \mathrm{K}$ and a cold reservoir at $300\ \mathrm{K}$.一台热机在 $500\ \mathrm{K}$ 热源与 $300\ \mathrm{K}$ 冷源之间运行。

(a) Calculate the maximum possible (Carnot) efficiency of any engine working between these reservoirs.计算在这两热源间运行的任何热机所能达到的最大（卡诺）效率。 [2]

(b) An inventor claims a real engine running between these same reservoirs has an efficiency of $45\%$. State, with a reason, whether the claim is possible.某发明者声称在这两热源间运行的真实热机效率为 $45\%$。说明该说法是否可能，并给出理由。 [2]

(c) State one change to the reservoir temperatures that would raise the Carnot ceiling, and explain briefly why it works.写出一种能提高卡诺上限的热源温度改动，并简要说明原因。 [2]

Graph and Data Questions图像与数据题

These items reward reading work as the area beneath a $p$-$V$ path and careful handling of signs and uncertainties. Quote uncertainties to one significant figure and round the value to match. For a monatomic ideal gas you may use $\Delta U = \tfrac{3}{2}\,n R\,\Delta T = \tfrac{3}{2}\,\Delta(pV)$.这些题考查把功读作 $p$-$V$ 路径下方面积，以及对符号与不确定度的细致处理。不确定度保留 1 位有效数字，并使数值末位与之对齐。对单原子理想气体可用 $\Delta U = \tfrac{3}{2}\,n R\,\Delta T = \tfrac{3}{2}\,\Delta(pV)$。

Q7HARD Paper 1B HL ONLY work as $p$-$V$ area (trapezium + isobaric)功即 $p$-$V$ 面积（梯形 + 等压） [10 marks]

A fixed mass of monatomic ideal gas follows a two-stage path on the $p$-$V$ diagram. In stage 1 (X to Y) the gas expands from $V = 1.0\times 10^{-3}\ \mathrm{m^{3}}$ to $4.0\times 10^{-3}\ \mathrm{m^{3}}$ while its pressure falls linearly along a straight line from $4.0\times 10^{5}\ \mathrm{Pa}$ to $1.0\times 10^{5}\ \mathrm{Pa}$. In stage 2 (Y to Z) the gas then expands at the constant pressure $1.0\times 10^{5}\ \mathrm{Pa}$ from $4.0\times 10^{-3}\ \mathrm{m^{3}}$ to $6.0\times 10^{-3}\ \mathrm{m^{3}}$.一定质量的单原子理想气体在 $p$-$V$ 图上沿两段路径变化。第 1 段（X 到 Y）：气体由 $V = 1.0\times 10^{-3}\ \mathrm{m^{3}}$ 膨胀到 $4.0\times 10^{-3}\ \mathrm{m^{3}}$，压强沿直线由 $4.0\times 10^{5}\ \mathrm{Pa}$ 线性降到 $1.0\times 10^{5}\ \mathrm{Pa}$。第 2 段（Y 到 Z）：气体在恒压 $1.0\times 10^{5}\ \mathrm{Pa}$ 下由 $4.0\times 10^{-3}\ \mathrm{m^{3}}$ 膨胀到 $6.0\times 10^{-3}\ \mathrm{m^{3}}$。

(a) Explain why, for stage 1, the work done by the gas cannot be found from $W = p\,\Delta V$ using a single pressure, and state what feature of the $p$-$V$ diagram gives the work instead.解释为何第 1 段不能用单一压强代入 $W = p\,\Delta V$ 求气体做的功，并说明 $p$-$V$ 图上的哪一特征给出功。 [2]

(b) Calculate the work done by the gas in stage 1 (X to Y) using the area of the trapezium beneath the straight-line path.用直线路径下方梯形的面积，计算第 1 段（X 到 Y）气体所做的功。 [3]

(d) For stage 2 only, calculate the change in internal energy of the monatomic gas using $\Delta U = \tfrac{3}{2}\,\Delta(pV)$, and hence the heat supplied in stage 2.仅对第 2 段，用 $\Delta U = \tfrac{3}{2}\,\Delta(pV)$ 计算单原子气体的内能变化，并由此求第 2 段供给的热量。 [3]

Q8HARD Paper 1B HL ONLY entropy bookkeeping + direction熵的核算与方向 [12 marks]

A student investigates the entropy change when $4200\ \mathrm{J}$ of heat is transferred between two large reservoirs whose temperatures stay constant. The table records the heat exchanged with each reservoir for a transfer from the hot reservoir ($700\ \mathrm{K}$) to the cold reservoir ($350\ \mathrm{K}$).一名学生研究 $4200\ \mathrm{J}$ 热量在两个温度恒定的大热源之间传递时的熵变。下表记录从热源（$700\ \mathrm{K}$）向冷源（$350\ \mathrm{K}$）传热时各热源交换的热量。

Reservoir热源	$T\ /\ \mathrm{K}$	$\Delta Q\ /\ \mathrm{J}$
Hot热源	$700$	$-4200$
Cold冷源	$350$	$+4200$

(a) Explain the signs of $\Delta Q$ in the table.解释表中 $\Delta Q$ 的符号。 [2]

(b) Calculate the entropy change of each reservoir, and hence the total entropy change of the two-reservoir system.计算每个热源的熵变，并由此求两热源系统的总熵变。 [4]

(c) Explain, using your result, why heat flows spontaneously from hot to cold but never the reverse, referring to the second law.结合所得结果并参照第二定律，解释为何热量自发地由热到冷流动而绝不反向。 [3]

(d) The cold reservoir's temperature is in fact known only to $\pm 10\ \mathrm{K}$. For the cold reservoir at $350\ \mathrm{K}$, calculate the percentage uncertainty in its entropy change.冷源温度实际上只精确到 $\pm 10\ \mathrm{K}$。对 $350\ \mathrm{K}$ 的冷源，计算其熵变的百分比不确定度。 [3]

Extended Structured Problems长结构问题

Set out each cycle clearly, stating the process type for each leg and the sign of $Q$ and $W$. Method marks dominate the longer items; carry intermediate values to extra figures and round only the final answer. For a monatomic ideal gas use $\Delta U = \tfrac{3}{2}\,\Delta(pV)$.把每个循环写清楚，标明每一段的过程类型及 $Q$、$W$ 的符号。长题中方法分占比最大；中间值多保留几位，仅在最终答案处取舍有效数字。对单原子理想气体用 $\Delta U = \tfrac{3}{2}\,\Delta(pV)$。

Q9HARD Paper 2 HL ONLY full $p$-$V$ cycle + efficiency完整 $p$-$V$ 循环与效率 [12 marks]

A fixed mass of monatomic ideal gas is taken once clockwise around the rectangular cycle A → B → C → D → A on a $p$-$V$ diagram, with the states:
A $(p = 2.0\times 10^{5}\ \mathrm{Pa},\ V = 2.0\times 10^{-3}\ \mathrm{m^{3}})$; B $(2.0\times 10^{5}\ \mathrm{Pa},\ 5.0\times 10^{-3}\ \mathrm{m^{3}})$; C $(1.0\times 10^{5}\ \mathrm{Pa},\ 5.0\times 10^{-3}\ \mathrm{m^{3}})$; D $(1.0\times 10^{5}\ \mathrm{Pa},\ 2.0\times 10^{-3}\ \mathrm{m^{3}})$. Legs A→B and C→D are isobaric; legs B→C and D→A are isochoric.一定质量的单原子理想气体沿 $p$-$V$ 图上的矩形循环 A → B → C → D → A 顺时针走一圈，各状态为：
A $(p = 2.0\times 10^{5}\ \mathrm{Pa},\ V = 2.0\times 10^{-3}\ \mathrm{m^{3}})$；B $(2.0\times 10^{5}\ \mathrm{Pa},\ 5.0\times 10^{-3}\ \mathrm{m^{3}})$；C $(1.0\times 10^{5}\ \mathrm{Pa},\ 5.0\times 10^{-3}\ \mathrm{m^{3}})$；D $(1.0\times 10^{5}\ \mathrm{Pa},\ 2.0\times 10^{-3}\ \mathrm{m^{3}})$。A→B 与 C→D 段为等压；B→C 与 D→A 段为等容。

(a) Calculate the work done by the gas on each of the four legs, and hence the net work done by the gas per cycle.计算四段中气体各自所做的功，并由此求每循环气体所做的净功。 [3]

(b) Show that the net work equals the area enclosed by the cycle on the $p$-$V$ diagram.证明净功等于 $p$-$V$ 图上循环所围的面积。 [2]

(c) For each leg calculate $\Delta U$ using $\Delta U = \tfrac{3}{2}\,\Delta(pV)$, and confirm that $\Delta U$ for the complete cycle is zero.用 $\Delta U = \tfrac{3}{2}\,\Delta(pV)$ 计算每段的 $\Delta U$，并确认整个循环的 $\Delta U$ 为零。 [3]

(d) Identify the legs on which heat enters the gas, calculate the total heat input $Q_{\mathrm{in}}$ per cycle, and hence find the thermal efficiency of the cycle.指出哪些段中热量进入气体，计算每循环的总吸热 $Q_{\mathrm{in}}$，并由此求该循环的热效率。 [4]

Q10HARD Paper 2 HL ONLY Carnot cycle + entropy + real engine卡诺循环、熵与真实热机 [10 marks]

A Carnot engine operates between a hot reservoir at $600\ \mathrm{K}$ and a cold reservoir at $300\ \mathrm{K}$. In each cycle it absorbs $1200\ \mathrm{J}$ of heat from the hot reservoir.一台卡诺热机在 $600\ \mathrm{K}$ 热源与 $300\ \mathrm{K}$ 冷源之间运行。每个循环它从热源吸收 $1200\ \mathrm{J}$ 热量。

(a) State the four processes that make up the Carnot cycle, in order.按顺序写出组成卡诺循环的四个过程。 [2]

(b) Calculate the efficiency of the engine, and hence the work output and the heat rejected to the cold reservoir per cycle.计算热机的效率，并由此求每循环的输出功与排向冷源的热量。 [4]

(c) Calculate the entropy change of the hot reservoir and of the cold reservoir per cycle, and comment on the total for this reversible engine.计算每循环热源与冷源的熵变，并就这台可逆热机的总熵变作出评论。 [3]

(d) A real engine between the same reservoirs has an efficiency of only $30\%$. State, with a reason, why its total entropy change per cycle is greater than that of the Carnot engine.在同样两热源间运行的真实热机效率仅为 $30\%$。说明为何其每循环的总熵变大于卡诺热机，并给出理由。 [1]

Thermodynamics热力学

Short Structured Items短结构题

Graph and Data Questions图像与数据题

Extended Structured Problems长结构问题