IB Physics HL | Unit E4: Fission

Show all working in the space below each question. Marks are awarded for correct method as well as final answers. Keep masses in $\mathrm{u}$ until the last step, then multiply by $931.5$ for energy in $\mathrm{MeV}$. Conserve nucleon number and proton number in every nuclear equation.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。把质量保持为 $\mathrm{u}$ 直到最后一步，再乘以 $931.5$ 得 $\mathrm{MeV}$ 能量。每个核方程都要守恒核子数与质子数。

Q1MEDIUM Paper 1 mass-energy equivalence质能等价 [4 marks]

In a nuclear process a mass of $0.20\ \mathrm{u}$ is converted entirely into energy.在某核过程中，$0.20\ \mathrm{u}$ 的质量被完全转化为能量。

(a) State the relation between mass and energy, and calculate the energy released in $\mathrm{MeV}$.写出质量与能量的关系，并计算释放的能量（以 $\mathrm{MeV}$ 计）。 [2]

(b) Express the same energy in joules, working from $1\ \mathrm{u} = 1.66\times 10^{-27}\ \mathrm{kg}$ and $c = 3.00\times 10^{8}\ \mathrm{m\,s^{-1}}$.由 $1\ \mathrm{u} = 1.66\times 10^{-27}\ \mathrm{kg}$ 与 $c = 3.00\times 10^{8}\ \mathrm{m\,s^{-1}}$ 出发，把同一能量以焦耳表示。 [2]

Q2MEDIUM Paper 1 mass defect and binding energy质量亏损与结合能 [6 marks]

A lithium-7 nucleus, ${}^{7}_{3}\mathrm{Li}$, has a measured nuclear mass of $7.01436\ \mathrm{u}$. Take $m_p = 1.00728\ \mathrm{u}$ and $m_n = 1.00867\ \mathrm{u}$.锂-7 核 ${}^{7}_{3}\mathrm{Li}$ 的实测核质量为 $7.01436\ \mathrm{u}$。取 $m_p = 1.00728\ \mathrm{u}$、$m_n = 1.00867\ \mathrm{u}$。

(a) Define the mass defect of a nucleus.给出核质量亏损的定义。 [1]

(b) Calculate the mass defect of lithium-7.计算锂-7 的质量亏损。 [2]

(c) Hence calculate the total binding energy, and the binding energy per nucleon, in $\mathrm{MeV}$.由此计算总结合能与比结合能（以 $\mathrm{MeV}$ 计）。 [3]

Q3HARD Paper 1 binding-energy-per-nucleon curve比结合能曲线 [6 marks]

The binding-energy-per-nucleon curve rises steeply for light nuclei, peaks near iron-56 at about $8.8\ \mathrm{MeV}$ per nucleon, then falls gently for heavy nuclei.比结合能曲线对轻核陡升，在铁-56 附近达峰，约 $8.8\ \mathrm{MeV}$ 每核子，随后对重核缓降。

(a) State what is meant by binding energy per nucleon, and write the expression for it in terms of mass defect.说明比结合能的含义，并写出其用质量亏损表示的表达式。 [2]

(b) Explain why iron-56, sitting at the peak of the curve, is the most stable nucleus and cannot release energy by either fission or fusion.解释为何位于曲线顶峰的铁-56 是最稳定的核，且无法通过裂变或聚变释放能量。 [2]

(c) State, in terms of the curve, the condition that must hold for a nuclear reaction to release energy.用曲线说明核反应释放能量必须满足的条件。 [2]

Q4HARD Paper 1 induced fission, balancing equations诱发裂变与配平方程 [6 marks]

A slow neutron is absorbed by a uranium-235 nucleus, which then undergoes induced fission according to ${}^{1}_{0}\mathrm{n} + {}^{235}_{\;92}\mathrm{U} \to {}^{140}_{\;54}\mathrm{Xe} + {}^{A}_{Z}\mathrm{Sr} + x\,{}^{1}_{0}\mathrm{n}$.一个慢中子被铀-235 核吸收，随后按 ${}^{1}_{0}\mathrm{n} + {}^{235}_{\;92}\mathrm{U} \to {}^{140}_{\;54}\mathrm{Xe} + {}^{A}_{Z}\mathrm{Sr} + x\,{}^{1}_{0}\mathrm{n}$ 发生诱发裂变。

(a) Explain what is meant by induced fission, and why a slow (thermal) neutron is used.解释诱发裂变的含义，以及为何使用慢（热）中子。 [2]

(b) By conserving nucleon number and proton number, determine $A$, $Z$ and the number of neutrons $x$.通过守恒核子数与质子数，求 $A$、$Z$ 及中子数 $x$。 [3]

Q5MEDIUM Paper 1 reactor components and enrichment反应堆部件与浓缩 [8 marks]

A thermal nuclear reactor is built from fuel, a moderator, control rods, a coolant, and containment shielding.热核反应堆由燃料、慢化剂、控制棒、冷却剂与安全壳屏蔽层构成。

(a) State, in one sentence each, the function of the moderator, the control rods, and the coolant.用一句话分别说明慢化剂、控制棒与冷却剂的作用。 [3]

(b) Explain what is meant by enrichment of the fuel, and why natural uranium must usually be enriched before use in such a reactor.解释燃料浓缩的含义，以及为何天然铀在用于此类反应堆前通常必须浓缩。 [3]

(c) State why the fission fragments produced in the fuel constitute radioactive nuclear waste.说明燃料中产生的裂变碎片为何构成放射性核废料。 [2]

Graph and Data Questions图像与数据题

These items reward careful reading of values off a curve and correct handling of uncertainties. Quote uncertainties to one significant figure and round the value to match. Carry intermediate figures and round only at the end.这些题考查从曲线上细致读值以及对不确定度的正确处理。不确定度保留 1 位有效数字，并使数值的末位与之对齐。中间值多保留几位，仅在最后取舍。

Q6HARD Paper 1B energy from the BE-per-nucleon curve由比结合能曲线求能量 [12 marks]

A uranium-236 nucleus (formed when ${}^{235}\mathrm{U}$ absorbs a neutron) fissions into barium-141 and krypton-92. The binding energy per nucleon of each nuclide is read from the curve below.铀-236 核（由 ${}^{235}\mathrm{U}$ 吸收中子形成）裂变为钡-141 与氪-92。各核素的比结合能从下表曲线读出。

Nuclide核素	${}^{236}\mathrm{U}$	${}^{141}\mathrm{Ba}$	${}^{92}\mathrm{Kr}$
$A$	$236$	$141$	$92$
$E_b / A\ /\ \mathrm{MeV}$	$7.6$	$8.3$	$8.5$

(a) State why the products of this fission lie higher on the binding-energy-per-nucleon curve than the reactant.说明这次裂变的产物为何在比结合能曲线上高于反应物。 [2]

(b) Calculate the total binding energy of the reactant nucleus and of the two product nuclei.计算反应物核与两块产物核的总结合能。 [3]

(d) Each value of $E_b / A$ is read from the curve with an absolute uncertainty of $\pm 0.1\ \mathrm{MeV}$. Calculate the percentage uncertainty in the value read for ${}^{236}\mathrm{U}$.每个 $E_b / A$ 值从曲线读出时的绝对不确定度为 $\pm 0.1\ \mathrm{MeV}$。计算 ${}^{236}\mathrm{U}$ 读数的百分比不确定度。 [2]

(e) State which form the released energy mainly takes immediately after the fission, and why.说明裂变后释放的能量主要以何种形式立即出现，以及原因。 [3]

Q7HARD Paper 1B reactor power from fission-rate data由裂变率数据求反应堆功率 [10 marks]

An operator logs the cumulative number of fissions in a reactor core at intervals during steady operation. Each fission releases $200\ \mathrm{MeV}$.操作员在稳定运行期间分时段记录堆芯累计裂变次数。每次裂变释放 $200\ \mathrm{MeV}$。

$t\ /\ \mathrm{s}$	$0$	$1.0$	$2.0$	$3.0$	$4.0$
cumulative fissions $/\ 10^{19}$累计裂变次数 $/\ 10^{19}$	$0$	$7.5$	$15.0$	$22.5$	$30.0$

(a) State what a graph of cumulative fissions against time would look like, and what its gradient represents.说明累计裂变次数对时间的图线会是什么形状，以及其斜率代表什么。 [2]

(b) Determine the fission rate (fissions per second).求裂变率（每秒裂变次数）。 [2]

(c) Convert $200\ \mathrm{MeV}$ to joules and hence calculate the thermal power output of the reactor.把 $200\ \mathrm{MeV}$ 转成焦耳，由此计算反应堆的热功率输出。 [3]

(d) State, with reference to the neutron multiplication factor $k$, why the cumulative-fission graph is a straight line rather than a rising curve.结合中子倍增因子 $k$，说明累计裂变图线为何是直线而非上升曲线。 [3]

Extended Structured Problems长结构问题

Set out each calculation clearly, stating the principle used. Method marks dominate the longer items; carry intermediate values to extra figures and round only the final answer. Keep masses in $\mathrm{u}$ until converting to energy.清晰列出每步计算并写明所用原理。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。把质量保持为 $\mathrm{u}$ 直到转成能量。

Q8HARD Paper 2 energy per fission from masses + power由质量求每次裂变能量与功率 [12 marks]

A uranium-235 nucleus captures a slow neutron and fissions: ${}^{1}_{0}\mathrm{n} + {}^{235}_{\;92}\mathrm{U} \to {}^{141}_{\;56}\mathrm{Ba} + {}^{92}_{36}\mathrm{Kr} + 3\,{}^{1}_{0}\mathrm{n}$. Use the atomic masses $m({}^{235}\mathrm{U}) = 235.04393\ \mathrm{u}$, $m({}^{141}\mathrm{Ba}) = 140.91440\ \mathrm{u}$, $m({}^{92}\mathrm{Kr}) = 91.92617\ \mathrm{u}$ and $m_n = 1.00867\ \mathrm{u}$.一个铀-235 核俘获一个慢中子并裂变：${}^{1}_{0}\mathrm{n} + {}^{235}_{\;92}\mathrm{U} \to {}^{141}_{\;56}\mathrm{Ba} + {}^{92}_{36}\mathrm{Kr} + 3\,{}^{1}_{0}\mathrm{n}$。取原子质量 $m({}^{235}\mathrm{U}) = 235.04393\ \mathrm{u}$、$m({}^{141}\mathrm{Ba}) = 140.91440\ \mathrm{u}$、$m({}^{92}\mathrm{Kr}) = 91.92617\ \mathrm{u}$、$m_n = 1.00867\ \mathrm{u}$。

(a) Verify that the equation conserves both nucleon number and proton number.验证该方程同时守恒核子数与质子数。 [2]

(b) Calculate the mass difference $\Delta m$ between the reactants and the products.计算反应物与产物之间的质量差 $\Delta m$。 [3]

(c) Hence calculate the energy released in this single fission, in $\mathrm{MeV}$ and in joules.由此计算这次单次裂变释放的能量，以 $\mathrm{MeV}$ 与焦耳表示。 [3]

(d) A reactor using this reaction has a thermal power output of $1.8\ \mathrm{GW}$. Calculate the number of fissions per second required.使用该反应的反应堆热功率输出为 $1.8\ \mathrm{GW}$。计算所需的每秒裂变次数。 [2]

(e) State why the electrical power delivered to the grid is significantly less than the thermal power output.说明输送到电网的电功率为何明显小于热功率输出。 [2]

Q9HARD Paper 2 chain reaction and critical mass链式反应与临界质量 [8 marks]

Each fission of uranium-235 releases on average $2.5$ fast neutrons. Whether the material sustains a chain reaction depends on the neutron multiplication factor $k$ and on the mass present.铀-235 每次裂变平均放出 $2.5$ 个快中子。材料能否维持链式反应取决于中子倍增因子 $k$ 及所含质量。

(a) Define the neutron multiplication factor $k$, and state its value for a reactor running at steady power.定义中子倍增因子 $k$，并写出反应堆稳定功率运行时的取值。 [2]

(b) Define critical mass, and explain in terms of surface-area-to-volume ratio why a sample below the critical mass cannot sustain a chain reaction.定义临界质量，并用表面积与体积之比解释为何低于临界质量的样品无法维持链式反应。 [4]

(c) Distinguish between the controlled chain reaction in a power reactor and the uncontrolled chain reaction in a fission weapon, in terms of $k$.用 $k$ 区分功率反应堆中的受控链式反应与裂变武器中的不受控链式反应。 [2]

Q10HARD Paper 2 reactor fuel consumption and waste反应堆燃料消耗与废料 [8 marks]

A power station has a reactor of thermal power output $3.0\ \mathrm{GW}$, with each fission releasing $200\ \mathrm{MeV}$. The overall efficiency of conversion to electrical power is $33\%$. Take the Avogadro constant as $6.02\times 10^{23}\ \mathrm{mol^{-1}}$ and the molar mass of uranium-235 as $235\ \mathrm{g\,mol^{-1}}$. One year $= 3.15\times 10^{7}\ \mathrm{s}$.某电站反应堆的热功率输出为 $3.0\ \mathrm{GW}$，每次裂变释放 $200\ \mathrm{MeV}$。转化为电功率的总效率为 $33\%$。取阿伏伽德罗常数 $6.02\times 10^{23}\ \mathrm{mol^{-1}}$，铀-235 摩尔质量 $235\ \mathrm{g\,mol^{-1}}$。一年 $= 3.15\times 10^{7}\ \mathrm{s}$。

(a) Calculate the electrical power delivered to the grid.计算输送到电网的电功率。 [1]

(b) Calculate the number of fissions per second needed to supply the thermal power.计算提供该热功率所需的每秒裂变次数。 [2]

(d) State one reason why the spent fuel removed from the reactor remains hazardous for a long time after the reactor is shut down.说明从反应堆中取出的乏燃料为何在停堆后长期仍具危害性的一个原因。 [2]

Fission核裂变

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