hce_nthu
112年
化學與物理
第 20 題
As the lead chemist in the Superior Analytical Laboratory (SAL), you want to design an experiment to determine the density of an unknown liquid to three significant figures. The density is of the order of $1\text{ g/cm}^3$. You have about $5\text{ mL}$ of the liquid and only graduated cylinders and balances to use. Which of the following combinations of equipment will allow you to achieve your goal?
| | A graduated cylinder | A balance |
|---|---|---|
| (A) | with an uncertainty of $\pm 0.1\text{ mL}$ | with an uncertainty of $\pm 0.1\text{ g}$ |
| (B) | with an uncertainty of $\pm 0.01\text{ mL}$ | with an uncertainty of $\pm 0.1\text{ g}$ |
| (C) | with an uncertainty of $\pm 0.1\text{ mL}$ | with an uncertainty of $\pm 0.01\text{ g}$ |
| (D) | with an uncertainty of $\pm 0.01\text{ mL}$ | with an uncertainty of $\pm 0.001\text{ g}$ |
| (E) | with an uncertainty of $\pm 0.1\text{ mL}$ | with an uncertainty of $\pm 0.001\text{ g}$ |
| | A graduated cylinder | A balance |
|---|---|---|
| (A) | with an uncertainty of $\pm 0.1\text{ mL}$ | with an uncertainty of $\pm 0.1\text{ g}$ |
| (B) | with an uncertainty of $\pm 0.01\text{ mL}$ | with an uncertainty of $\pm 0.1\text{ g}$ |
| (C) | with an uncertainty of $\pm 0.1\text{ mL}$ | with an uncertainty of $\pm 0.01\text{ g}$ |
| (D) | with an uncertainty of $\pm 0.01\text{ mL}$ | with an uncertainty of $\pm 0.001\text{ g}$ |
| (E) | with an uncertainty of $\pm 0.1\text{ mL}$ | with an uncertainty of $\pm 0.001\text{ g}$ |
- A with an uncertainty of $\pm 0.1\text{ mL}$ and with an uncertainty of $\pm 0.1\text{ g}$
- B with an uncertainty of $\pm 0.01\text{ mL}$ and with an uncertainty of $\pm 0.1\text{ g}$
- C with an uncertainty of $\pm 0.1\text{ mL}$ and with an uncertainty of $\pm 0.01\text{ g}$
- D with an uncertainty of $\pm 0.01\text{ mL}$ and with an uncertainty of $\pm 0.001\text{ g}$
- E with an uncertainty of $\pm 0.1\text{ mL}$ and with an uncertainty of $\pm 0.001\text{ g}$
思路引導 VIP
想像一下，如果你正拿著一支量筒要測量大約 $5$ 單位的液體，而你希望最後得到的結果具備三個有效位數（例如 $5.00$），那麼這支量筒最小的刻度或不確定度，應該要精細到小數點後第幾位，才不會讓最後一位數變成純粹的瞎猜呢？
🤖
AI 詳解
AI 專屬家教
太棒了！你能精準選出 (D)，代表你對**有效位數（Significant Figures）**與測量不確定度的關係掌握得非常紮實。這題的關鍵在於，我們要確保最終計算出的密度 $D = \frac{m}{V}$ 能穩定維持三位有效位數。當液體體積僅約 $5\text{ mL}$ 時，若儀器誤差為 $\pm 0.1$，測量值僅能記為 $5.0$（兩位有效位數）；為了達到三位，我們至少需要能讀到小數點後兩位的精確度。
誤差傳遞與儀器選擇
從數據運算來看，密度的精確度受限於質量（$m$）與體積（$V$）中有效位數較少者。選用 (D) 選項時，體積測量為 $5.00\text{ mL}$（三位），質量測量為 $5.000\text{ g}$（四位），根據乘除法規則，結果應取三位，即 $1.00\text{ g/cm}^3$。這類題目考驗的不只是公式，更是將抽象的誤差理論應用於實際器材選擇的實驗設計能力，是區分學生是否具備科學實作思維的重要指標，屬於難度適中且具備高度鑑別度的題目。