A-Level Physics Unit 5 Jan 2019 Formula Derivations | A-Level 物理 2019年1月单元5公式推导

📚 A-Level Physics Unit 5 Jan 2019 Formula Derivations | A-Level 物理 2019年1月单元5公式推导

In A-Level Physics Unit 5, students explore thermodynamics, nuclear decay, oscillations and gravitational fields. The January 2019 past paper examined several core derivations that are essential for a deep understanding of these topics. This article revisits the key formula derivations that frequently appear, providing clear step-by-step explanations to aid revision.

在A-Level物理单元5中，学生们学习热力学、核衰变、振动和引力场。2019年1月的试卷考查了若干核心公式推导，对于深入理解这些专题至关重要。本文重温常考的关键公式推导，提供清晰的逐步解释，以辅助复习。

1. Pressure of an Ideal Gas | 理想气体的压强推导

The kinetic theory models gas pressure as the result of countless collisions of molecules with the container walls. Consider a cube of side L containing N molecules, each of mass m. A single molecule moving with velocity component v_x perpendicular to a wall will undergo a change in momentum of 2mv_x upon elastic collision. The time between successive collisions with the same wall is 2L / v_x, so the average force exerted by one molecule is (2mv_x) / (2L/v_x) = mv_x all molecules and accounting for three dimensions, the total force on a wall is (N / 3) m〈v²〉/ L. Since pressure p = Force / Area = Force / L², we obtain p = ⅓ (N / V) m〈c²〉, where V = L³ and 〈c²〉 is the mean square speed. This can also be written as pV = ⅓ N m〈c²〉.

气体动理论把气体压强看作分子与容器壁无数碰撞的结果。考虑一个边长为 L 的立方体，内有 N 个分子，每个分子质量为 m。一个分子以垂直于壁面的速度分量 v_x 运动，弹性碰撞后动量变化为 2mv_x。与同一壁面两次碰撞的时间间隔为 2L/v_x，因此一个分子施加的平均力为 (2mv_x)/(2L/v_x) = mv_x²/L。对所有分子求和并考虑三维运动，壁面上的总力为 (N/3) m〈v²〉/L。压强 p = 力/面积 = 力/L²，于是得到 p = ⅓ (N/V) m〈c²〉，其中 V = L³，〈c²〉是均方速率。也可写为 pV = ⅓ N m〈c²〉。

pV = ⅓ N m〈c²〉

2. Ideal Gas Equation | 理想气体状态方程推导

Combining the kinetic theory equation pV = ⅓ N m〈c²〉 king temperature to molecular kinetic energy. The total kinetic energy of N molecules is E_k = ½ N m〈c²〉. Substituting m〈c²〉 = 2E_k / N gives pV = ⅔ E_k. For one mole (n=1, N = N_A), experiment shows pV = RT, therefore RT = ⅔ E_k (per mole), implying E_k = (3/2) RT. For n moles, E_k r ideal gas equation pV = nRT, which unifies the macroscopic and microscopic descriptions.

将气体动理论的 pV = ⅓ N m〈c²〉与经验理想气体定律 pV = nRT 结合，需要将温度与分子动能联系起来。N 个分子的总动能 E_k = ½ N m〈c²〉。代入 m〈c²〉= 2E_k/N 得到 pV = ⅔ E_k。对一摩尔 (n=1，N=N_A)，实验给出 pV = RT，故 RT = ⅔ E_k（每摩尔），从而 E_k = (3/2) RT。对 n 摩尔，E_k = (3/2) nRT，最终得到常见的理想气体方程 pV = nRT，统一了宏观与微观的描述。

pV = nRT

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3. Mean Kinetic Energy and Temperature | 平均动能与温度的关系

From the above, the average translational kinetic energy per molecule is ½ m〈c²〉 = (3/2) kT, where k = R / N_A is Boltzmann's constant. This derivation shows that temperature is a direct measure of the random kinetic energy of particles. It also gives a practical formula for the root-mean-square speed: c_rms = √3RT / M, where M is the molar mass. This relationship is pivotal in explaining rates of diffusion and effusion.

由上述推导，每个分子的平均平动动能 ½ m〈c²〉 = (3/2) kT，其中 k = R / N_A 为玻尔兹曼常数。该推导表明温度是粒子无序动能的直接量度。同时也给出方均根速率的实用公式：c_rms = √3RT/M，M 为摩尔质量。这一关系在解释扩散和隙流速率时至关重要。

½ m〈c²〉 = (3/2) kT

c_rms = √3RT / M

4. Radioactive Decay Law | 放射性衰变定律推导

f N is the number of undecayed nuclei, the activity A = -dN/dt = λN. Separating variables:

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