Internal Energy
A-Level Physics Topic 16 15:27 English narration · English + 中文 subtitles burned in
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Pump up a bicycle tyre, quickly and hard, and something surprising happens: the pump gets hot.
又快又用力地给自行车轮胎打气，会发生一件令人惊讶的事：打气筒变热了。
Warm enough to feel.
热得能感觉到。
You did not heat it with a flame.
你并没有用火去加热它。
So where did the warmth come from?
那这份热从哪里来？
From you.
来自你自己。
As you pushed the piston down, you did work on the air inside, squeezing it.
当你把活塞往下压时， 你对里面的空气做了功，把它压缩。
That work went straight into the air's internal energy, making its molecules move faster — and faster molecules mean a higher temperature.
那份功直接进入了空气的内能，让它的分子运动得更快—— 而更快的分子意味着更高的温度。
This is the first law of thermodynamics, in your hands.
这就是热力学第一定律，就在你手中。
Energy inside matter has a name, and a rulebook.
物质内部的能量有一个名字，也有一套规则。
Today: internal energy, the work a gas can do, and the first law of thermodynamics that ties heat and work together.
今天：内能、气体能做的功， 以及把热和功联系起来的热力学第一定律。
Let's begin.
让我们开始吧。
Everything is made of jiggling, wandering molecules — and they carry energy.
一切都由颤动、游走的分子构成——而它们携带着能量。
The total energy of all of them, added up, is the internal energy of the object.
把它们所有的能量加在一起， 就是这个物体的内能。
It comes in two parts: the kinetic energy of their motion, and the potential energy of the forces between them.
它由两部分组成：分子运动的动能，以及分子之间作用力的势能。
It depends only on the current state — not on how the object got there.
它只取决于当前的状态——而与物体是怎么到达这个状态的无关。
Heat something up, and its internal energy rises.
给它加热，它的内能就升高。
Look closer at that kinetic energy, because molecules move in more than one way.
再仔细看看这份动能，因为分子的运动方式不止一种。
Every molecule flies through space — that is translational motion, and a single atom, like helium, has only this.
每个分子都在空间中飞行——这就是平动， 像氦这样的单原子只有这一种运动。
A molecule of two or more atoms can also spin, which is rotational motion, and its bonds can stretch and squash, which is vibrational motion.
由两个或更多原子组成的分子还能自转，这就是转动， 它的化学键还能伸长和压缩，这就是振动。
All three are random, and all three count towards the internal energy.
这三种运动都是随机的，都要算进内能里。
Add the potential energy stored in the forces between neighbouring molecules, and you have both halves of U.
再加上相邻分子之间作用力储存的势能，你就得到了内能的两个部分。
For a real solid, liquid or gas, both parts matter: the molecules sit close enough to pull on each other, so the potential energy is genuinely there.
对真实的固体、液体或气体来说，两部分都重要：分子彼此靠得足够近，会互相吸引， 所以势能是真实存在的。
But the ideal-gas model makes one big simplification — it ignores the forces between molecules altogether.
但理想气体模型做了一个很大的简化——它完全忽略分子之间的作用力。
No forces means no intermolecular potential energy, so that whole term is zero, and the internal energy of an ideal gas is purely kinetic.
没有作用力就没有分子间势能，所以那一整项等于零，理想气体的内能就全部是动能。
That is why, for an ideal gas, internal energy comes down to a single question: how fast are the molecules moving?
这就是为什么对理想气体来说，内能只归结为一个问题：分子运动得有多快？
Two points examiners love.
有两点是考官最爱考的。
First, internal energy depends only on the state of the system — its temperature, pressure, volume and amount of substance.
第一，内能只取决于系统的状态——它的温度、压强、体积和物质的量。
Two identical gases in the same state have the same internal energy, no matter what path they took to get there.
两份处于相同状态的相同气体，内能就相同，不管它们是通过什么途径到达这个状态的。
Heat then compress, or compress then heat: same U.
先加热再压缩，或者先压缩再加热：内能一样。
Second, internal energy is a sum over the molecules, not the kinetic energy of the whole object moving.
第二，内能是对所有分子求和， 而不是整个物体运动的动能。
A cylinder of gas on a speeding train has bulk kinetic energy, but that is separate from U, which counts only the random motion inside.
一瓶气体放在飞驰的火车上，它有整体的动能， 但那与内能无关；内能只计算里面的随机运动。
Zoom in on a gas, and you see internal energy directly.
放大来看气体，你就直接看到了内能。
The molecules dart about at tremendous speeds, colliding and rebounding, never stopping.
分子以惊人的速度四处乱窜，碰撞、反弹，永不停歇。
That ceaseless motion is kinetic energy — and for an ideal gas, it is the whole of the internal energy.
这不停的运动就是动能——而对理想气体来说，它就是全部的内能。
The hotter the gas, the faster they move.
气体越热，它们运动得越快。
Temperature is just a measure of this hidden, molecular energy.
温度不过是这份隐藏的、分子的能量的一种量度。
Let's make that exact.
我们把它写精确。
From Topic fifteen, every molecule of an ideal gas has an average translational kinetic energy of three-halves k T, where k is the Boltzmann constant.
由第十五章可知，理想气体每个分子的平均平移动能是二分之三 k T， 其中 k 是玻尔兹曼常量。
Multiply by N molecules, and since the potential energy is zero, that sum is the entire internal energy: U equals three-halves N k T, which is also three-halves n R T.
乘上 N 个分子，又因为势能为零， 这个总和就是全部的内能：U 等于二分之三 N k T，也等于二分之三 n R T。
Read what that says.
看看这说明了什么。
Internal energy is directly proportional to the thermodynamic temperature — the temperature in kelvin.
内能与热力学温度成正比——也就是以开尔文为单位的温度。
Double the temperature and you double the internal energy.
温度加倍，内能也加倍。
Careful though: this clean proportionality is exact only for an ideal gas.
但要小心：这么干净的正比关系只对理想气体才严格成立。
Here is the case that catches people out.
有一种情况最容易让人出错。
Melt ice, or boil water, and while it changes phase the temperature does not move at all — it sits at zero degrees, or at one hundred.
把冰熔化，或者把水烧开，在相变的过程中温度完全不动—— 它就停在零度，或者停在一百度。
Yet you are pouring energy in.
可是你一直在往里输入能量。
So where does the energy go?
那么能量去哪儿了？
Not into the kinetic term, because the temperature is flat.
不是进入动能那一项，因为温度是平的。
It goes into the potential term: the energy drags the molecules apart against the forces holding them together, breaking bonds.
它进入了势能那一项： 这份能量把分子拉开，克服把它们束在一起的作用力，把键打断。
So during a phase change the internal energy rises while the temperature stays constant — and that is only possible because U has a potential part.
所以在相变过程中，温度不变而内能上升——而这只有在内能包含势能部分时才可能。
A gas can do work.
气体能做功。
Let it expand, and it pushes a piston outward.
让它膨胀，它就把活塞向外推。
At constant pressure, the work done is simply the pressure, times the change in volume.
在恒定压强下，所做的功就等于压强乘以体积的变化。
But be careful about direction.
但要小心方向。
When a gas expands, it does work on its surroundings — it gives energy out.
当气体膨胀时，它对周围环境做功——把能量放出去。
When it is compressed, the surroundings do work on it — energy goes in.
当它被压缩时，周围环境对它做功——能量进来。
Same equation, opposite sign.
同一个公式，符号相反。
Where does that formula come from?
这个公式是怎么来的？
Force times distance, nothing more.
就是力乘距离，没有别的。
The gas presses on a piston of cross-sectional area A.
气体压在横截面积为 A 的活塞上。
Pressure is force per unit area, so the force on the piston is p times A.
压强是单位面积上的力，所以活塞上受到的力是 p 乘 A。
Now let the piston move out a small distance delta x.
现在让活塞向外移动一小段距离 delta x。
Work is force times distance, so the work done is p A delta x.
功等于力乘距离，所以做的功是 p 乘 A 乘 delta x。
And look at A times delta x: that is exactly the volume swept out by the piston, which is the increase in the gas's volume, delta V.
再看 A 乘 delta x：这正是活塞扫过的体积，也就是气体体积的增加量 delta V。
So the work done is p delta V, provided the pressure stays constant while the piston moves.
所以做的功就是 p delta V——前提是活塞移动时压强保持不变。
This is how a steam turbine works, and with it a whole power station: expanding steam pushes the blades around.
蒸汽轮机就是这样工作的，整座发电厂也是：膨胀的蒸汽推动叶片转动。
Numbers, then.
来算数字。
A gas at a constant pressure of one times ten to the five pascals expands from two times ten to the minus three cubic metres to five times ten to the minus three cubic metres.
一份气体在一乘十的五次方帕的恒定压强下， 从二乘十的负三次方立方米膨胀到五乘十的负三次方立方米。
Find the work done by the gas.
求气体做的功。
Pause and try it.
先暂停，自己算一算。
Ready?
好了吗？
First, the volume change: five minus two, times ten to the minus three, is three times ten to the minus three cubic metres.
先算体积变化：五减二，再乘十的负三次方， 等于三乘十的负三次方立方米。
Then the work: one times ten to the five, multiplied by three times ten to the minus three, gives three hundred joules of work done by the gas on its surroundings.
再算功：一乘十的五次方，乘以三乘十的负三次方， 得到三百焦耳——这是气体对周围环境做的功。
There is a picture that makes this obvious.
有一张图能让这件事变得一目了然。
Put pressure up the vertical axis and volume across the bottom.
把压强画在竖轴上，把体积画在横轴上。
An expansion at constant pressure is then a horizontal line, running from V one to V two.
恒压下的膨胀就是一条水平线，从 V 一延伸到 V 二。
The work done is the area under that line — and the area of a rectangle is height times width, which is p times delta V. Exactly our formula.
所做的功就是这条线下面的面积—— 而矩形的面积等于高乘宽，也就是 p 乘 delta V，正好就是我们的公式。
The picture is stronger than the formula, because it still works when the pressure changes: the work is always the area under the curve.
这张图比公式更有力，因为压强变化时它照样成立：功永远等于曲线下的面积。
And at constant volume the line is vertical, with no area at all, so no work is done.
而在等容过程中，图线是竖直的，根本没有面积，所以不做功。
Now the sign rule, where marks are won and lost.
现在来看符号规则，分数就在这里得失。
This syllabus always writes the first law with W meaning the work done on the gas.
本考纲写第一定律时，W 一律指外界对气体做的功。
So compress the gas and delta V is negative: the work done on it is positive, and the gas gains energy.
所以压缩气体时 delta V 是负的：对气体做的功是正的，气体获得能量。
Let the gas expand and delta V is positive: the work done on it is negative, because now the gas is giving energy out.
让气体膨胀时 delta V 是正的：对气体做的功是负的，因为现在是气体把能量放出去。
Read the wording of the question.
要看清题目的措辞。
Work done on the gas is positive when it is compressed; work done by the gas is the opposite sign, positive when it expands.
"对气体做的功"在压缩时为正；"气体做的功"符号相反，在膨胀时为正。
Same size, opposite sign.
大小相同，符号相反。
And if the volume never changes, no work is done at all.
而如果体积始终不变，就完全不做功。
Now the great bookkeeping rule: the first law of thermodynamics.
现在来看那条伟大的记账规则：热力学第一定律。
The change in a system's internal energy equals the heat added to it, plus the work done on it.
一个系统内能的变化， 等于加给它的热量，加上对它做的功。
It is simply energy is conserved when heat and work pass between a system and its surroundings — conservation of energy, for heat.
它其实就是关于热的能量守恒。
Warm a gas, and its internal energy climbs.
给气体加热，它的内能上升。
Compression does work on it, and it climbs again.
压缩它，内能又上升。
Every joule is accounted for — nothing is ever lost.
每一焦耳都算得清清楚楚——没有任何东西会丢失。
Let's use the law.
我们来用这条定律。
A gas absorbs five hundred joules of heat while it expands, doing two hundred joules of work on its surroundings.
一份气体在膨胀的同时吸收了五百焦耳的热量， 并对周围环境做了两百焦耳的功。
Find the change in its internal energy.
求它内能的变化。
Pause it there.
先暂停一下。
The trap is the sign.
陷阱在符号上。
The gas does the work, so the work done on the gas is minus two hundred joules, while q is plus five hundred.
是气体在做功，所以外界对气体做的功是负两百焦耳，而 q 是正五百焦耳。
Add them: delta U equals five hundred plus minus two hundred, which is three hundred joules.
加起来：delta U 等于五百加负两百，也就是三百焦耳。
The internal energy rises by three hundred joules — and for an ideal gas, that means the temperature rose too.
内能上升了三百焦耳—— 而对理想气体来说，这意味着温度也升高了。
Read the equation once more, because it hides something useful.
再读一遍这个方程，因为它藏着一件很有用的事。
Delta U is fixed by the change of state — for an ideal gas, purely by the change in temperature.
delta U 由状态的变化决定—— 对理想气体来说，只由温度的变化决定。
But q and W can trade off in any mixture that adds up to it.
但 q 和 W 可以以任何加起来相同的组合互相替换。
All heat and no work: that is heating a gas in a sealed rigid container, and delta U equals q.
全靠加热、不做功：那就是在密闭刚性容器里给气体加热，delta U 等于 q。
All work and no heat: that is compressing a gas in a perfectly insulated cylinder, and delta U equals W.
全靠做功、不加热：那就是在完全绝热的气缸里压缩气体，delta U 等于 W。
Two completely different experiments, the same rise in internal energy, the same final temperature.
两个完全不同的实验，内能的升高相同，最终温度也相同。
Four processes come up again and again, and one graph holds all of them.
有四种过程会反复出现，而一张图就能装下它们全部。
Draw each one from the same starting state, this point here.
让每一种都从同一个起始状态出发， 就是这个点。
A horizontal line is constant pressure.
水平线是等压过程。
A vertical line is constant volume.
竖直线是等容过程。
A curve falling away to the right with the temperature held fixed is isothermal.
向右下方落下、温度保持不变的曲线是等温过程。
A steeper curve below it, with no heat crossing the boundary, is adiabatic.
在它下面更陡的那条、没有热量穿过边界的曲线，是绝热过程。
Before we take them one at a time, hold on to the anchor: for an ideal gas, delta U is three-halves n R delta T.
在逐个分析之前，先记住这个锚点：对理想气体，delta U 等于二分之三 n R delta T。
Internal energy tracks temperature and nothing else, whichever path the gas takes.
不管气体走哪条路径，内能只跟着温度走。
Take the two curves first, and read each one with the first law.
先看两条曲线，并用第一定律来读它们。
Isothermal means the temperature is constant, so delta T is zero and delta U is zero.
等温意味着温度不变，所以 delta T 为零，delta U 也为零。
The law becomes zero equals q plus W, so q equals minus W: any heat that flows in comes straight back out as work done by the gas.
定律变成零等于 q 加 W，也就是 q 等于负 W：流进去的热量又原样以气体做功的形式流出来。
Adiabatic means no heat flows at all, so q is zero and delta U equals W.
绝热意味着完全没有热量流动，所以 q 为零，delta U 等于 W。
Work is the only way in or out — squeeze the gas and it warms, let it expand and it cools.
功是唯一的进出通道——压缩气体它就变热，让它膨胀它就变冷。
That is the bicycle pump, and the spray can, exactly.
这正是自行车打气筒，也正是喷雾罐。
Now the two straight lines.
再看两条直线。
Constant volume means a sealed, rigid container: the gas cannot move anything, so no work is done, so q equals delta U.
等容意味着一个密闭的刚性容器：气体推不动任何东西， 所以不做功，于是 q 等于 delta U。
Every joule you supply becomes internal energy, and the temperature climbs fastest here.
你供给的每一焦耳都变成内能，这里温度升得最快。
Constant pressure means the same gas under a piston that is free to slide.
等压意味着同样的气体上面盖着一个可以自由滑动的活塞。
Heat it and it expands, pushing the piston out and doing work on the surroundings, so the work done on the gas is minus p delta V.
给它加热，它就膨胀，把活塞推出去，对周围环境做功，所以对气体做的功是负 p delta V。
Your heat now has two jobs to pay for: raising the internal energy, and doing the expansion work.
这时你的热量要付两笔账：既要提高内能，又要完成膨胀做的功。
Same heat in, smaller temperature rise.
供给同样的热量，温度升高得更少。
That constant-volume case gives a result worth memorising.
等容这种情况给出一个值得记住的结果。
No work is done, so the heat supplied equals the change in internal energy, which for an ideal gas is three-halves n R delta T.
不做功，所以供给的热量就等于内能的变化， 对理想气体来说等于二分之三 n R delta T。
So warming one mole by one kelvin at constant volume costs three-halves R of energy: about twelve point five joules.
于是在等容条件下把一摩尔气体升温一开尔文，需要二分之三 R 的能量：约十二点五焦耳。
That is the molar heat capacity at constant volume of a monatomic ideal gas.
这就是单原子理想气体的等容摩尔热容。
You are not required to use the symbol C V, but you are expected to produce q equals three-halves n R delta T for constant-volume heating.
考纲不要求你使用符号 C V， 但要求你能写出等容加热时 q 等于二分之三 n R delta T。
One last example, the kind that looks worse than it is.
最后一道例题，它看起来比实际更难。
A sample of ideal gas starts at temperature T with internal energy U.
一份理想气体最初处于温度 T，内能为 U。
In step one it is compressed until its temperature is three T, and work W is done on the gas.
第一步，它被压缩到温度变成三 T，外界对它做了功 W。
In step two it is cooled at constant volume down to two T.
第二步，它在体积不变的条件下被冷却到二 T。
Find the heat transfer in each step.
求每一步的热量传递。
Step one.
先看第一步。
For an ideal gas internal energy is proportional to temperature, so at three T it is three U, and delta U one is two U.
对理想气体，内能与温度成正比，所以在三 T 时内能是三 U，delta U 一等于二 U。
The work done on the gas is plus W, so q one is two U minus W.
外界对气体做的功是正 W，所以 q 一等于二 U 减 W。
Step two is cooling from three T to two T at constant volume.
第二步是在体积不变的条件下从三 T 冷却到二 T。
The internal energy falls from three U to two U, so delta U two is minus U.
内能从三 U 降到二 U， 所以 delta U 二等于负 U。
Constant volume means no work, so W two is zero, and the first law gives q two is minus U.
等容意味着不做功，所以 W 二等于零， 由第一定律得 q 二等于负 U。
It is negative, which tells you heat flowed out of the gas — exactly what cooling means.
它是负的，这告诉你热量流出了气体——冷却正是这个意思。
Now check the whole journey.
现在检查整个过程。
The total change in internal energy is two U minus U, which is plus U; and the gas went from T to two T, so U should have become two U, a rise of U.
内能的总变化是二 U 减 U，等于正 U； 而气体从 T 变到二 T，内能本该从 U 变成二 U，也就是升高 U。
It agrees.
两者一致。
One last case, and it ties everything together.
最后一种情况，它把所有内容串在一起。
Boil water in an open pan, at constant atmospheric pressure.
在敞口锅里把水烧开，压强是恒定的大气压。
You supply a lot of heat, so q is large and positive, yet the temperature never climbs above one hundred degrees.
你供给大量的热，所以 q 又大又正，可是温度始终不会超过一百度。
Two things swallow that energy.
有两件事吞掉了这些能量。
The latent heat drags the molecules apart against their attractions, raising the potential part of the internal energy.
潜热克服分子之间的吸引把它们拉开， 提高了内能中的势能部分。
And the steam takes up far more room than the water did, so the gas expands and does work on the atmosphere, making W negative.
而水蒸气占的空间比水大得多， 所以气体膨胀并对大气做功，使 W 为负。
The first law still holds, term for term: delta U equals q plus W.
第一定律依然逐项成立：delta U 等于 q 加 W。
Let's see it in action.
我们来看它的实际作用。
Compress a gas fast, doing work on it, and with no time for heat to escape, all that work becomes internal energy — the gas heats up.
快速压缩气体，对它做功，而热量来不及逃走，那份功就全变成内能—— 气体变热。
That is the bicycle pump.
这就是自行车打气筒。
Reverse it: let a gas expand and do work, and its internal energy falls — the gas cools down.
反过来：让气体膨胀并做功，它的内能下降——气体变冷。
That is why a spray can turns cold as you use it.
这就是为什么喷雾罐用着用着会变凉。
Work in, hotter; work out, colder.
功进来，变热；功出去，变冷。
Three marks to secure.
三个要拿稳的分。
First, internal energy is the sum of the random kinetic and potential energies of the molecules.
第一，内能是分子随机的动能和势能之和。
Second, use the first law: change in internal energy equals heat in, plus work done on the system — mind the signs.
第二，用第一定律：内能的变化等于进来的热量，加上对系统做的功——注意符号。
Third, at constant pressure, work equals pressure times the change in volume.
第三，在恒定压强下，功等于压强乘以体积的变化。
Master these, and thermodynamics is yours.
掌握这些，热力学就是你的了。