Internal energy · 内能
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| internal energy/ɪnˈtɜːnl ˈenədʒi/ | 内能 | nèi néng |
| translational/trænˈsleɪʃənl/ | 平动 | píng dòng |
| rotational/rəʊˈteɪʃənl/ | 转动 | zhuǎn dòng |
| vibrational/vaɪˈbreɪʃənl/ | 振动 | zhèn dòng |
| ideal gas/aɪˈdɪəl ɡæs/ | 理想气体 | lǐ xiǎng qì tǐ |
| intermolecular/ˌɪntəməˈlekjʊlə/ | 分子间 | fèn zǐ jiān |
A cup of tea that is going nowhere, fast
- A cup of tea sits still on the table. Nothing about it is moving, and yet its molecules are travelling at hundreds of metres per second, in every direction at once.
- The velocities cancel, which is why the cup stays put. The energies do not cancel, because kinetic energy has no direction to cancel with.
- That energy, invisible from outside, is what makes the tea hot and what a kettle had to pay for.
- This lesson is internal energy 内能: what it is made of, what it depends on, and what happens to it when a temperature does not change.
一杯哪儿也不去、却飞快的茶
- 一杯茶静静地放在桌上。它没有任何部分在移动,可它的分子正以每秒几百米的速度、同时朝各个方向飞行。
- 速度互相抵消,这就是杯子不动的原因。能量不会抵消,因为动能没有方向可以抵消。
- 这份从外面看不见的能量,正是让茶变热的东西,也是水壶必须付出的代价。
- 这一课讲内能(internal energy):它由什么构成、取决于什么,以及温度不变时它会怎样。
What internal energy is made of
- The internal energy $U$ of a system is the sum of the random distribution of the kinetic and potential energies of its molecules.
- The kinetic part: molecules fly through space (translational 平动), and unless they are single atoms they also spin (rotational 转动) and shake (vibrational 振动).
- The potential part: the energy stored in the forces between molecules, which depends on how far apart they are.
- Both marks of the two-mark definition need "sum of kinetic and potential energies" and the word random. Leaving out "random" describes the energy of a moving object instead.
内能由什么构成
- 一个系统的内能 $U$ 是其分子的动能和势能的无规则分布之和。
- 动能部分:分子在空间中飞行(平动,translational),而且除非它们是单原子的,它们还会自转(转动,rotational)和振动(振动,vibrational)。
- 势能部分:储存在分子之间的作用力中的能量,它取决于分子相距多远。
- 两分定义的两分需要"动能与势能之和"以及"无规则"这个词。漏掉"无规则",描述的就成了一个运动物体的能量。
The spread of molecular energies · 分子能量的分布
Internal energy is the total random kinetic + potential energy of the molecules. Heat the gas and the whole speed distribution shifts to higher energy. · 内能是分子随机动能与势能的总和。加热气体,整个速率分布就向更高能量移动。
Internal energy is the sum of which energies? · 内能是哪些能量的总和?
Internal energy = random molecular KE + intermolecular PE. The object's overall motion (bulk KE) is separate. · 内能 = 随机分子动能 + 分子间势能。物体的整体运动(整体动能)是分开的。
Which is the full two-mark definition of internal energy? · 哪一个是内能的完整两分定义?
Both marks need "kinetic and potential" and the word "random". Without "random" the phrase describes a moving object's energy instead. · 两分都需要"动能与势能"和"无规则"这个词。没有"无规则",这句话描述的就成了运动物体的能量。
Random, not bulk
- $U$ is a sum over the molecules, not the kinetic energy of the object moving as a whole.
- A tanker of gas driving down a motorway has a great deal of bulk kinetic energy. Its internal energy is exactly the same as when it was parked, because the random molecular motion has not changed.
- Stop the tanker and the bulk kinetic energy becomes heat, which then does raise $U$. The two are separate quantities that can be converted into one another.
One of these has internal energy; both have energy
是无规则的,不是整体的
- $U$ 是对分子求和,不是物体作为整体运动的动能。
- 一辆在高速公路上行驶的气体槽车有大量的整体动能。它的内能与它停在那里时完全相同,因为无规则的分子运动没有改变。
- 让槽车停下,整体动能变成热,那时它才会提高 $U$。两者是可以互相转化的、分开的量。
其中一个有内能;两个都有能量
A moving train's bulk kinetic energy counts as part of its internal energy. · 一列运动火车的整体动能算作它内能的一部分。
No — internal energy is the energy of the random molecular motion, not the whole object moving along. · 不——内能是随机分子运动的能量,不是整个物体的运动。
A tanker of gas driving along a motorway has more internal energy than the same tanker parked, at the same temperature. · 在高速公路上行驶的气体槽车,其内能比同温度下停放的同一辆槽车更大。
Bulk motion is not internal energy. The random molecular motion is unchanged, so U is the same; the lorry simply also has kinetic energy. · 整体运动不是内能。无规则分子运动没有改变,所以 U 相同;那辆车只是另外还有动能而已。
A state function
- $U$ is determined by the state of the system: its temperature, pressure, volume and amount of substance.
- It does not depend on the path taken to reach that state. Compress a gas and then heat it, or heat it and then compress it: if the final state is the same, $U$ is the same.
- That is what makes the first law useful in the next lesson: $\Delta U$ can be found from the endpoints alone, however complicated the journey.
状态函数
- $U$ 由系统的状态决定:它的温度、压强、体积和物质的量。
- 它不取决于到达那个状态所走的路径。先压缩气体再加热,或先加热再压缩:只要末态相同,$U$ 就相同。
- 这正是下一课里第一定律好用的原因:无论过程多复杂,$\Delta U$ 都能只由两端求出。
Internal energy depends only on the state of the system, not on the path taken to reach it. · 内能只取决于系统的状态,而不取决于到达该状态所经过的路径。
Yes — $U$ is a function of state (T, p, V, amount); two routes to the same state give the same $U$. · 是的——$U$ 是状态的函数(T、p、V、量);到达同一状态的两条路径给出相同的 $U$。
Temperature and internal energy
- Raising an object's temperature raises the random kinetic energy of its molecules, and so raises its internal energy.
- For an ideal gas 理想气体 the intermolecular 分子间 forces are ignored, so the molecular potential energy is zero and the internal energy is purely kinetic. With $\tfrac32 kT$ per molecule:
- So for an ideal gas $U$ is directly proportional to the thermodynamic temperature. Double $T$ and you double $U$. This is exact only for an ideal gas.
温度与内能
- 提高物体的温度会提高其分子的无规则动能,因而提高它的内能。
- 对理想气体(ideal gas),分子间(intermolecular)作用力被忽略,所以分子势能为零,内能全部是动能。每个分子 $\tfrac32 kT$:
- 所以对理想气体,$U$ 与热力学温度成正比。$T$ 加倍,$U$ 也加倍。这只对理想气体精确成立。
For an ideal gas, the internal energy is: · 对于理想气体,内能是:
No intermolecular PE, so $U$ is all kinetic: $U = \tfrac{3}{2}NkT = \tfrac{3}{2}nRT$. · 没有分子间势能,所以 $U$ 全是动能:$U = \tfrac{3}{2}NkT = \tfrac{3}{2}nRT$。
For an ideal gas, if the absolute temperature doubles, the internal energy multiplies by: · 对于理想气体,如果绝对温度加倍,内能变为原来的多少倍:
$U = \tfrac{3}{2}nRT \propto T$, so doubling $T$ doubles $U$. · $U = \tfrac{3}{2}nRT \propto T$,所以 $T$ 加倍使 $U$ 加倍。
Worked example: a gas heated at constant pressure
- Sketch how the internal energy of a fixed mass of ideal gas varies with its volume as it is heated at constant pressure.
- At constant pressure, Charles's law gives $V \propto T$. For an ideal gas, $U \propto T$.
- Therefore $U \propto V$: a straight line through the origin.
- The origin is on the line because at absolute zero both the extrapolated volume and the internal energy are zero. Say why the line passes through the origin; that is usually the second mark.
例题:恒压加热的气体
- 画出固定质量的理想气体在恒压加热时,内能随体积变化的图。
- 恒压下,查理定律给出 $V \propto T$。对理想气体,$U \propto T$。
- 因此 $U \propto V$:一条过原点的直线。
- 原点在这条线上,因为在绝对零度时外推的体积和内能都为零。要说出这条线为什么过原点;那通常是第二分。
A fixed mass of ideal gas is heated at constant pressure. What does a graph of internal energy against volume look like? · 固定质量的理想气体被恒压加热。内能对体积的图像是什么样子?
At constant pressure V is proportional to T, and for an ideal gas U is proportional to T, so U is proportional to V. It passes through the origin because both extrapolate to zero at absolute zero. · 恒压下 V 正比于 T,而对理想气体 U 正比于 T,所以 U 正比于 V。它过原点,因为两者在绝对零度处外推都为零。
Phase changes
- When ice melts or water boils, the temperature does not change, yet energy is still being supplied.
- The kinetic energy is unchanged, because the temperature is unchanged. The potential energy rises, as the bonds between molecules are broken and their separation grows.
- So $U$ increases at constant temperature, by the latent heat supplied. This is the clearest case where internal energy and temperature come apart.
相变
- 冰融化或水沸腾时,温度不变,可是仍在不断供给能量。
- 动能不变,因为温度不变。势能上升,因为分子之间的键被打断、间距变大。
- 所以 $U$ 在温度不变的情况下增加,增加量就是供给的潜热。这是内能与温度分道扬镳的最清楚的例子。
While water boils at constant temperature, its internal energy: · 当水在恒温下沸腾时,它的内能:
The temperature (and so KE) is unchanged, but energy goes into breaking bonds — raising the molecular PE, so $U$ rises. · 温度(以及动能)不变,但能量用于破坏键——提高分子势能,所以 $U$ 上升。
Worked example: name both energies, every time
- A three-mark "describe and explain, with reference to molecular kinetic and potential energies" answer says what happens to each:
- A gas heated at constant volume: the kinetic energy increases, because temperature measures mean molecular kinetic energy; the potential energy is unchanged, since the separation of the molecules does not change; so $U$ increases.
- A wire stretched within its elastic limit at constant temperature: the kinetic energy is unchanged, same temperature; the potential energy increases, because the atoms are pulled further apart against the interatomic forces; so $U$ increases.
- Ice melting at $0\ ^\circ\text{C}$: the kinetic energy is unchanged; the potential energy increases as bonds break and separations grow; so $U$ increases by the latent heat supplied.
例题:每一次都把两种能量都说出来
- 三分的"结合分子动能与势能描述并解释"的答案要说清每一种的变化:
- 恒容加热的气体:动能增加,因为温度是分子平均动能的量度;势能不变,因为分子的间距没有改变;所以 $U$ 增加。
- 在弹性极限内于恒温下被拉伸的金属丝:动能不变,温度相同;势能增加,因为原子被拉得更开、克服了原子间作用力;所以 $U$ 增加。
- $0\ ^\circ\text{C}$ 下融化的冰:动能不变;势能增加,因为键被打断、间距变大;所以 $U$ 增加,增量就是供给的潜热。
Match each change to what happens to the molecular kinetic and potential energies. · 把每种变化与分子动能和势能的变化配对。
Temperature changes the kinetic part; changing the separation of the molecules changes the potential part. Name both in every answer. · 温度改变动能部分;改变分子间距改变势能部分。每个答案都要把两者都说出来。
Marks that slip away
- The definition needs random and both energies. "The energy of the molecules" is not enough.
- Bulk motion is not internal energy. A moving object's kinetic energy is separate from $U$.
- $U = \tfrac32 nRT$ holds for an ideal gas only, because only there is the molecular potential energy zero.
- At a phase change the temperature is constant but $U$ rises, through the potential term. Say which energy changes and which does not.
容易丢掉的分
- 定义需要"无规则"和两种能量。"分子的能量"不够。
- 整体运动不是内能。运动物体的动能与 $U$ 是分开的。
- $U = \tfrac32 nRT$ 只对理想气体成立,因为只有那里分子势能才为零。
- 相变时温度不变而 $U$ 上升,靠的是势能那一项。要说清哪种能量变了、哪种没变。
You've got it
- internal energy is the sum of the random distribution of the kinetic and potential energies of a system's molecules
- it is a property of the state, not of the path, and it is separate from the bulk kinetic energy of a moving object
- for an ideal gas the potential term is zero, so $U = \tfrac32 nRT$ and $U \propto T$ exactly
- a phase change raises $U$ at constant temperature by raising the potential energy as bonds break
你掌握了
- 内能是系统分子的动能与势能的无规则分布之和
- 它是状态的属性而不是路径的,并且与运动物体的整体动能是分开的
- 对理想气体势能项为零,所以 $U = \tfrac32 nRT$,$U$ 精确正比于 $T$
- 相变通过提高势能(键被打断)在温度不变时提高 $U$