Maxwell–Boltzmann Energy Distribution & Reaction Kinetics Suite

Interactive Controls

Set & Compare Conditions

Tick to overlay

Condition A (Reference / Baseline) Shade Area A (E ≥ E_a,1)

Temperature (T₁) 300 K

Activation Energy (E_a,1) 5.0 kJ/mol

Condition B (Variable T & Catalyst) Shade Area B (E ≥ E_a,2)

Catalyst State:

Temperature (T₂)

500 K

Activation Energy (E_a,2) 3.0 kJ/mol (Catalyzed)

Area & Particle Count Comparison

Total Particles (N) 1,000,000

Reactive (Cond A) 0

Reactive (Cond B) 0

Loading comparison ratio...

Interactive Observation Guide:

Effect of Temperature (T): Increasing T flattens and shifts the curve to the right, but the total area under the curve remains constant at N.
Effect of Catalyst (E_a): Adding a catalyst lowers E_a to E_a', shifting the threshold line leftward without altering the distribution curve shape.
Compare Overlays: Toggle the interactive catalyst state or temperature match buttons in Condition B to compare reactive areas under identical or different conditions.

HKDSE Answering Tips / 應試答題技巧

Effect of temperature:

An increase in temperature can increase the average kinetic energy of the molecules.
There are more molecules having kinetic energy equal to or greater than the activation energy.
This will increase the effective collision frequency.

Effect of catalyst:

When a catalyst is added, the reaction would proceed in an alternative pathway with lower activation energy.
For the pathway with lower activation energy, more particles would possess sufficient kinetic energy to react than that in the original pathway with higher activation energy as the frequency of effective collisions is greater.

Advanced Explanation (Out of Syllabus) / 詳細理論與問答彙整

Complete bilingual theoretical breakdown and step-by-step mathematical derivations (Sections 0 – 12).

# Maxwell–Boltzmann 曲線下面積為何代表粒子總數？ / Why does the area under a Maxwell–Boltzmann curve represent the total number of particles?

0. 核心結論 / Core conclusion

中文： Maxwell–Boltzmann 圖的 y-軸嚴格來說通常不是「分子數」，而是「每單位速率的分子數」或「每單位能量的分子數」，也就是一種密度 (density)。所以：

小面積 = 高度 \times 寬度

才代表某一小範圍內的分子數。若曲線是 normalized probability density function，即機率密度函數 f(v)，總面積是 1。若曲線乘上總粒子數 N，即 Nf(v)，總面積就是 N。

English: Strictly speaking, the y-axis of a Maxwell–Boltzmann graph is usually not simply “number of molecules.” It is “number of molecules per unit speed” or “number of molecules per unit energy,” i.e. a density. Therefore:

small area = height \times width

represents the number of molecules in a small range. If the curve is a normalized probability density function f(v), the total area is 1. If the curve is multiplied by the total number of particles N, i.e. Nf(v), the total area is N.

1. Probability density function 是甚麼？ / What is a probability density function?

中文： 先想「擲骰仔」。骰仔結果是 discrete，不連續的：只可能是 1, 2, 3, 4, 5, 6。所以我們可以問：

P (X = 3) = 16

但分子速率 v 是 continuous，連續的。理論上 v 可以是 300.0000 m/s、300.0001 m/s、300.000001 m/s……有無限多個可能值。因此問「剛好等於 300 m/s 的機率」沒有實際意義；對連續變量，單一精確值的機率是 0。我們要問的是「速率落在一個範圍內」的機率，例如 300 至 310 m/s。NIST 的統計手冊也用積分面積來定義連續機率分佈，並指出總面積要等於 1。

English: Think first about rolling a die. A die outcome is discrete: it can only be 1, 2, 3, 4, 5, 6. So we can ask:

P (X = 3) = 16

But molecular speed v is continuous. In theory, v could be 300.0000 m/s, 300.0001 m/s, 300.000001 m/s, and so on. There are infinitely many possible values. Therefore, asking for the probability of exactly 300 m/s is not useful; for a continuous variable, the probability of one exact value is 0. Instead, we ask for the probability that the speed lies in an interval, such as 300 to 310 m/s. NIST defines continuous probability distributions through area/integrals and states that the total area must be 1.

2. PDF 的定義 / Definition of a PDF

中文： 若 X 是連續變量，它的 probability density function，簡稱 PDF，寫作 f_X(x)。它滿足：

P (a \leq X \leq b) = \int a b f X (x) d x

以及：

f X (x) \geq 0, \int -\infty \infty f X (x) d x = 1

重點是：f_X(x) 本身不是機率；面積才是機率。

English: If X is a continuous variable, its probability density function, or PDF, is written as f_X(x). It satisfies:

P (a \leq X \leq b) = \int a b f X (x) d x

and:

f X (x) \geq 0, \int -\infty \infty f X (x) d x = 1

The key point is: f_X(x) itself is not a probability; area under the curve is probability.

3. 用 histogram 理解 PDF / Understanding PDF through a histogram

中文： 想像你量度 N 個分子的速率，然後分箱： 0–100 m/s、100–200 m/s、200–300 m/s。如果某個速率區間內有 N_i 個分子，而區間寬度是 Δv，你可以畫一條 histogram bar：

bar height = N i Δ v &implies; bar area = N i Δ v \times Δ v = N i

當 Δv 越來越細，histogram 就變成平滑曲線。這條曲線的高度不是「分子數」，而是

dNdv

，即「每單位速率的分子數」。

English: Imagine measuring the speeds of N molecules and grouping them into bins (0–100 m/s, 100–200 m/s, etc.). If a bin contains N_i molecules and has width Δv, the histogram height can be written as:

bar height = N i Δ v &implies; bar area = N i Δ v \times Δ v = N i

As Δv becomes very small, the histogram approaches a smooth curve. The height of the curve is not directly “number of molecules,” but

dNdv

, meaning “number of molecules per unit speed.”

4. Maxwell–Boltzmann 速率分佈 / Maxwell–Boltzmann speed distribution

中文： 對理想氣體，在熱平衡下，Maxwell–Boltzmann 速率分佈通常寫成：

f speed (v) = 4π \cdot (m 2π k B T) 3/2 \cdot v 2 e - m v 2 2 k B T

其中 v 是速率 (v ≥ 0)，m 是單個分子的質量，T 是絕對溫度，k_B 是 Boltzmann constant，f_speed(v) 是速率的 probability density function。OpenStax 也用這個分佈來描述理想氣體分子速率，並定義 dN = N f(v) dv，即 v 到 v + dv 之間的分子數。

English: For an ideal gas at thermal equilibrium, the Maxwell–Boltzmann speed distribution is usually written as above, where v ≥ 0 is speed, m is mass of one molecule, T is absolute temperature, k_B is Boltzmann constant, and f_speed(v) is the PDF of speed. OpenStax uses this distribution for ideal-gas molecular speeds and defines dN = N f(v) dv.

5. 為何總面積是 1 或 N？ / Why is the total area 1 or N?

中文： 如果 f_speed(v) 是 PDF，它已被 normalized：∫₀^∞ f_speed(v) dv = 1。這代表「所有可能速率的機率加起來是 100%」。若氣體樣本中有 N 個分子，則真正的「分子數密度」是 n(v) = N f_speed(v)。整個曲線下面積就是：

\int 0 \infty n (v) d v = \int 0 \infty N f speed (v) d v = N

PDF 曲線下面積 = 1。
分子數密度曲線 N f(v) 下面積 = N。

English: If f_speed(v) is a PDF, it is normalized to 1. If the gas sample contains N molecules, the actual number-density curve is n(v) = N f_speed(v). The total area under the whole number-density curve is N.

6. 單位分析：為何 y-axis 不應只寫「分子數」？ / Unit analysis

中文： 如果 x-axis 是速率 v，單位是 m/s。若曲線下面積要代表分子數，則 y-axis 必須有單位：

moleculesm/s

，因為

moleculesm/s

× m/s = molecules。如果 x-axis 是能量 E，單位是 joule (J)，則 y-axis 應該是

moleculesJ

，因為

moleculesJ

× J = molecules。所以很多教科書寫 y-axis 為 “number of molecules” 是簡化說法；嚴格應寫 “number of molecules per unit speed” 或 “number of molecules per unit energy”。

English: If the x-axis is speed v (m/s), the y-axis unit must be molecules/(m/s). If the x-axis is energy E (J), the y-axis unit should be molecules/J. Labeling the y-axis simply as "number of molecules" is a simplification.

7. Maxwell–Boltzmann 分佈的簡單推導 / Simple derivation

在熱平衡下，Boltzmann factor 給出 P_i ∝ e^{- ε_i / (k_B T)} 或 P_i =

e^{- ε_i / (k_B T)}Q

。

Step 1: 單粒子動能 / Kinetic energy of one particle: E =

m (v_x² + v_y² + v_z²) &implies; P ∝ e^{- m(v_x² + v_y² + v_z²)2 k_B T}

Step 2: 歸一化 / Normalization: f_v(v_x, v_y, v_z) = (

m2π k_B T

)^3/2 e^{- m(v_x² + v_y² + v_z²)2 k_B T}，其三維空間全域積分等於 1。

Step 3: 由 velocity vector 變成 speed distribution: 速率介於 v 與 v + dv 的速度向量在速度空間構成一球殼，體積為 4π v² dv。故 f_speed(v) = 4π (

m2π k_B T

)^3/2 v² e^{- m v²2 k_B T}。

8. 能量分佈版本 / Energy distribution version

中文： 很多化學圖不是畫 speed，而是畫 kinetic energy E =

m v²。將 speed distribution 轉成 energy distribution：

f E (E) = 2 \sqrtπ \cdot \sqrt E (k B T) 3/2 \cdot e - E k B T (E \geq 0)

若有 N 個分子：n_E(E) = N f_E(E)，那麼能量在 E 到 E + dE 的分子數是： dN = n_E(E) dE = N f_E(E) dE。

English: Transform speed distribution to kinetic energy distribution using E =

m v². The particle density curve is n_E(E) = N f_E(E), and dN = n_E(E) dE gives molecules between E and E + dE.

9. 為何 activation energy 右邊的面積代表可反應分子比例？ / Activation energy

中文： 在化學反應圖中，通常有一條 activation energy E_a。若 x-axis 是 kinetic energy，則能量大於 E_a 的分子比例是：

fraction with E ≥ Ea = ∫Ea∞ fE(E) dE

如果用的是分子數密度 n_E(E) = N f_E(E)，則能量大於 E_a 的分子數是 ∫_{E_a}^∞ n_E(E) dE，比例即為

∫_{E_a}^∞ n_E(E) dEN

。

English: In kinetic energy graphs, the area to the right of E_a represents the fraction or exact number of molecules with E ≥ E_a. Adding a catalyst lowers E_a to E_a', expanding this reactive area without altering the shape of the curve.

10. 面積不是總能量 / The area is not total energy

中文： 這一點很重要。若 x-axis 是 energy，曲線 n_E(E) 的小面積 n_E(E) dE 代表該能量範圍內的分子數，不是能量。若要計算總能量，需要額外乘以 E：

total kinetic energy = \int 0 \infty E \cdot n E (E) d E

所以 ∫₀^∞ n_E(E) dE = N，但 ∫₀^∞ E · n_E(E) dE 才是總動能。

English: Area under n_E(E) represents particle count N. Total energy requires integrating E · n_E(E) dE.

11. 最常見誤解 / Common misunderstandings

「y-axis 是分子數」不夠準確： 應該是分子數密度，例如
dNdv
或
dNdE
。
PDF 的高度不是機率： 機率是面積 f(v) dv。
單一精確速率的機率是 0： 不代表沒有分子，連續變量需考慮範圍。
PDF 總面積是 1，不是 N： 只有 N f(v) 的總面積才是 N。
同一批氣體在不同溫度下，總分子數不變，所以面積不變： 高溫時曲線變矮、變闊、峰值向右移；但若 N 不變，總面積仍是 N。

Maxwell–Boltzmann Interactive Learning Suite