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化學 1889

離子的電動勢作用

瓦爾特·能斯特

濃度之差,悄然就是一份電壓。

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In depth · the introduction

取同樣的鹽水,調成濃淡兩份,各插一片金屬,電壓表的指針就會一動——濃度之差,悄然就是一份電壓。

核心想法

電池能發電,是因為化學物質「想」從擁擠的地方挪向稀疏的地方。能斯特的成就,是說清了這份「想」究竟值多少電壓。他的方程式,把一個電池的電壓,與其中離子的濃度聯繫了起來。

由此落下的經驗法則,簡潔得漂亮:在室溫下,單價離子濃度每差十倍,約值 59 毫伏。一側濃十倍:59 mV。濃一百倍:約 118 mV。電壓隨濃度比的對數增長,而非隨比值本身。

它是如何誕生的

1880 年代末,全新的物理化學,正在萊比錫奧斯特瓦爾德的實驗室裡被一磚一瓦地建起來。兩個想法剛剛抵達那裡:范托夫已表明,溶解的顆粒像氣體一樣向外施壓(滲透壓);阿倫尼烏斯則主張,水中的鹽早已裂成帶電的離子。瓦爾特·能斯特,一位敏銳而雄心勃勃的年輕助手,看出了如何把二者熔在一起。

他推想:如果一個離子像受壓的氣體,那麼一片金屬電極就同時感受到兩股相反的推力——它自身溶成離子的傾向,與水中已有離子推回來的壓強。二者平衡之處,定下電壓。在 1889 年的特許任教論文裡,他把這幅圖景寫成了一道方程式。(他會在 1920 年拿到諾貝爾獎,但那是為了完全不同的東西——他的熱定理,即熱力學第三定律。)

它為何重要

在能斯特之前,你能按金屬反應的劇烈程度排個序,卻無法預言一份電壓。在他之後,你能把它算出來。正是這一項本領,支撐著電池的設計、用 pH 計測量酸度、讀出血樣中鈉或鈣含量的感測器,以及「金屬為何會腐蝕」的科學——還有,驚人地,你體內每一個神經細胞跨膜維持的那一份微小電壓。

一個可以想像的畫面

想像相鄰的兩個房間,由一道門相連,一間擠滿了人,另一間幾乎空著。人們自然會從擁擠的一間往空的一間飄移;若在門口裝一道旋轉閘,這股飄移就能把它轉動、做功。人群差越大,閘轉得越費力。一個離子濃度電池正是如此:離子「想」從濃的一側挪向稀的一側,而這份「想」,就以電壓的形式顯現。能斯特方程式,正是「擁擠」與「伏特」之間那個精確的兌換率。

兩個盛有同種鹽、濃度不同的燒杯由鹽橋相連,配一只電壓表;拖動濃度與溫度滑桿、選擇離子電荷,看電池電壓隨濃度差增大而上升、在兩側相等時歸零。

它的位置

伏打在 1800 年造出第一個電池,法拉第在 1830 年代把電解量化,但二者都說不出,為什麼某個電池會給出某個特定的電壓。能斯特補上了這一環,並把它架在范托夫與阿倫尼烏斯關於溶液的新想法之上。從這裡,一條線徑直通向霍奇金與赫胥黎對神經衝動的解釋——它建立在同一道方程式之上——也通向你此刻讀著這段文字的設備裡的電池。

The original document
Original source text
Walther Nernst · Zeitschrift für physikalische Chemie 4 (1889): 129–181 · Leipzig
Written as Nernst's habilitation in Wilhelm Ostwald's Leipzig laboratory — the same rooms where van 't Hoff's osmotic theory and Arrhenius's ions were turning physical chemistry into a quantitative science — the memoir asks a deceptively simple question: how large is the electrical force an ion can exert, and on what does it depend?
1 · The osmotic analogy
Nernst takes over van 't Hoff's result that dissolved particles behave like a gas, exerting an osmotic pressure proportional to their concentration, and Arrhenius's claim that in solution a salt is already split into free ions. An ion in solution is therefore like a gas under pressure; differences in that pressure are differences in a tendency to move.
2 · Electrolytic solution pressure (Lösungstension)
To this he adds one new quantity: every metal is supposed to have an “electrolytic solution pressure” P, an intrinsic tendency to shed ions into the solution, leaving electrons behind on the metal. Against it pushes the osmotic pressure p of the ions already dissolved, which tends to drive them back onto the metal. The electrode comes to rest where the two are balanced, and the charge separation built up in reaching that balance is the electrode potential.
3 · The resulting law
Equating the electrical work of moving the ions against the osmotic work of compressing them from one pressure to the other gives a logarithmic law: the potential depends on the logarithm of the concentration (strictly, the ratio of solution pressure to osmotic pressure). In the form used ever since, the electromotive force of a cell is E = E° − (RT/nF) ln Q.
E = (RT / nF) · ln(P / p) → E = E° − (RT / nF) · ln Q ; 2.303 RT/F ≈ 59.2 mV per tenfold concentration ratio at 25 °C.
4 · Concentration cells and consequences
The clearest test is a cell with the same metal in the same salt at two concentrations: the standard term cancels and a voltage appears from the concentration difference alone. From the same law Nernst reads off how cell voltages, solubilities and equilibria depend on concentration — the working equations of electrochemistry.
[ … ]
The literal picture of a “solution pressure” was later dropped in favour of Gibbs's chemical potential, and dilute concentrations were replaced by activities — but the logarithmic law itself, and the 59 mV-per-decade slope, are exactly as Nernst left them.
Leipzig · 1889