📚 IB Physics: Calculating Average Kinetic Energy of Gas Molecules | IB物理：气体分子平均动能计算
In the IB Physics syllabus, the kinetic theory of gases is a foundational topic that links the microscopic world of moving molecules to macroscopic observables such as pressure and temperature. The formula for the average kinetic energy of gas molecules is central to this topic. Mastering its derivation and application will help you solve a wide range of exam problems involving molecular motion, internal energy, and root-mean-square speed.
在 IB 物理课程中，气体分子动理论是连接微观分子运动与宏观可观测物理量（如压强和温度）的基础课题。气体分子平均动能的公式是这一部分的核心。掌握其推导与应用，将帮助你解决大量涉及分子运动、内能以及方均根速度的考试题目。
1. Ideal Gas Assumptions | 理想气体假设
In the ideal gas model, gas molecules are treated as point masses with no volume. Collisions between molecules and with the container walls are perfectly elastic. There are no inter-molecular forces except during collisions, and all molecules move in random directions with a range of speeds.
在理想气体模型中，气体分子被看作是没有体积的质点。分子之间以及与容器壁之间的碰撞是完全弹性的。除碰撞瞬间外，分子间不存在相互作用力，所有分子都沿随机方向运动，且速率各不相同。
2. Deriving the Pressure Formula | 压强公式的推导
ll is Δt = 2L / vₓ. Hence the average force exerted on the wall is F = Δp / Δt = mvₓ² / L.
考虑一个边长为 L 的立方体容器，其中有一个质量为 m、沿 x 方向运动且速度分量为 vₓ 的分子。该分子撞击右壁时，其动量变化为 Δp = -2mvₓ。连续两次与同一面壁碰撞的时间间隔为 Δt = 2L / vₓ，因此该分子对壁的平均作用力为 F = Δp / Δt = mvₓ² / L。
For N molecules, the total pressure is p = N m ⟨vₓ²⟩ / V. For random motion, the three velocity components contribute equally, so ⟨vₓ²⟩ = ⟨vᵧ²⟩ = ⟨vz²⟩ = (1/3)⟨v²⟩. Substituting this into the pressure expression gives:
对于 N 个分子，总压强为 p = N m ⟨vₓ²⟩ / V。由于分子做无规则运动，三个速度分量对压强的贡献相同，因此 ⟨vₓ²⟩ = ⟨vᵧ²⟩ = ⟨vz²⟩ = (1/3)⟨v²⟩。将这一关系代入压强的表达式，可得：
pV = (1/3) N m ⟨v²⟩
3. Expressing Average Kinetic Energy | 平均动能的表示
The average translational kinetic energy of a single molecule is defined as ⟨Ek⟩ = (1/2) m ⟨v²⟩. Using the pressure formula from Section 2, we can rewrite pV as:
单个分子的平均平动动能定义为 ⟨Ek⟩ = (1/2) m ⟨v²⟩。利用第 2 节的压强公式，我们可以把 pV 改写成：
pV = (2/3) N ⟨Ek⟩
4. Connecting to Temperature | 与温度的联系
The ideal gas law for N molecules can be written as pV = N kB T, where kB = 1.38 × 10⁻²³ J/K is the Boltzmann constant. Equating this with pV = (2/3) N ⟨Ek⟩ gives:
对于 N 个分子的理想气体状态方程可写为 pV = N kB T，其中 kB = 1.38 × 10⁻²³ J/K 为玻尔兹曼常数。将其与 pV = (2/3) N ⟨Ek⟩ 联立，可得：
⟨Ek⟩ = (3/2) kB T
This simple result shows that the average kinetic energy of a gas molecule depends only on the absolute temperature, not on the mass or type of gas. This is why temperature is a measure of the average kinetic energy of gas particles.
这一简洁的结论表明，气体分子的平均动能只取决于绝对温度，而与气体的种类或分子质量无关。这正是“温度是气体分子平均动能大小的量度”这一说法的由来。
5. Root-Mean-Square Speed | 方均根速度
Using ⟨Ek⟩ = (1/2) m ⟨v²⟩ and ⟨Ek⟩ = (3/2) kB T, the root-mean-square speed vrms is:
利用 ⟨Ek⟩ = (1/2) m ⟨v²⟩ 和 ⟨Ek⟩ = (3/2) kB T，可得方均根速率 vrms 为：
vrms = √3 kB T / m
When expressed with molar mass M, this becomes vrms = √3RT / M where R = 8.31 J/(mol·K). Note that m must be in kilograms when using the molecular form.
若使用摩尔质量 M，则可写成 vrms = √3RT / M，其中 R = 8.31 J/(mol·K)。注意使用分子质量形式时，m 必须以千克为单位。
6. Degrees of Freedom and Internal Energy | 自由度与内能
The result ⟨Ek⟩ = (3/2) kB al motion. For monatomic gases, the total internal energy of the system is U = N × (3/2) kB reedom, giving U = (5/2) nRT. For polyatomic gases the internal energy can be even higher.
⟨Ek⟩ = (3/2) kB T 只考虑了分子的平动动能。对于单原子气体，系统的总内能 U = N × (3/2) kB T = (3/2) nRT，其中 n 为摩尔数。双原子气体在常温下还有两个转动自由度，因此 U = (5/2) nRT。对于多原子气体，内能还会更高。
7. Worked Examples | 例题解析
Example 1. For helium
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