📚 IB Physics: Derivation of Kinetic Theory Pressure Formula (Formula 396) | IB 物理：Physics for the IB Diploma 396 公式推导（气体动理论压强公式）
In the IB Physics course, particularly using the textbook ‘Physics for the IB Diploma’ by K.A. Tsokos, students encounter a series of derivations labelled by formula numbers. Formula 396 corresponds to the key result from the kinetic theory of gases: P = ⅓ ρ ⟨c²⟩. This derivation connects microscopic particle motion to macroscopic pressure and is fundamental to understanding ideal gases.
在IB物理课程中，特别是使用K.A. Tsokos编写的《Physics for the IB Diploma》教材时，学生会遇到一系列由公式编号标记的推导。公式396对应的是气体动理论的关键结果：P = ⅓ ρ ⟨c²⟩。这一推导将微观粒子运动与宏观压强联系起来，是理解理想气体的基础。
1. Assumptions of the Kinetic Theory | 动理论的基本假设
The kinetic theory model makes several simplifying assumptions about an ideal gas in order to apply Newtonian mechanics to a huge number of particles.
动理论模型对理想气体做出若干简化假设，以便将牛顿力学应用于数目巨大的粒子系统。
The gas consists of a large number of identical, tiny particles (molecules) in constant, random motion.
气体由大量相同的微小粒子（分子）组成，它们处于持续无规运动中。
The volume of the molecules themselves is negligible compared to the volume of the container.
与容器的体积相比，分子本身的体积可忽略不计。
All collisions between molecules and with the walls of the container are perfectly elastic.
所有分子间以及分子与器壁之间的碰撞均为完全弹性碰撞。
There are no intermolecular forces except during collisions; between collisions, molecules move in straight lines at constant speed.
除碰撞瞬间外，分子间不存在相互作用力；在两次碰撞之间，分子以恒定速率沿直线运动。
The duration of a collision is negligible compared to the time between collisions.
碰撞的持续时间与两次碰撞之间的时间间隔相比可以忽略。
2. Setting Up the Derivation | 推导模型的建立
To begin the derivation, we consider a cubic container of side length L containing N molecules, each of mass m. We focus on one molecule moving with velocity components v_x, v_y, v_z.
为了开始推导，我们考虑一个边长为 L 的立方体容器，内有 N 个分子，每个分子质量为 m。我们关注其中一个分子，其速度分量为 v_x、v_y、v_z。
The molecule collides with the wall perpendicular to the x-axis. After an elastic collision, its x-velocity reverses from +v_x to –v_x, while the y– and z-components remain unchanged.
该分子与垂直于 x 轴的器壁碰撞。经过弹性碰撞后，其 x 方向速度由 +v_x 变为 –v_x，而 y 和 z 方向的分量保持不变。
The change in momentum of the molecule in the x-direction is Δp_x = m(–v_x) – m(v_x) = –2m v_x. By Newton’s third law, the wall experiences an equal and opposite change in momentum: +2m v_x.
分子在 x 方向上的动量变化为 Δp_x = m(–v_x) – m(v_x) = –2m v_x。根据牛顿第三定律，器壁受到等大反向的动量变化：+2m v_x。
3. Time Between Collisions with the Same Wall | 与同一器壁两次碰撞的时间间隔
After rebounding, the molecule travels to the opposite wall and back. The distance covered along the x-direction between successive collisions with the same wall is 2L.
反弹后，分子将运动到对面的器壁再返回。与同一器壁连续两次碰撞之间，分子沿 x 方向移动的距离为 2L。
The time interval between such collisions is therefore Δt = 2L / v_x, assuming no collisions with other molecules interfere.
因此，连续两次碰撞的时间间隔为 Δt = 2L / v_x，这里假设不受与其他分子碰撞的干扰。
The average force exerted by this one molecule on the wall is given by the rate of change of momentum: F = Δp / Δt = (2m v_x) / (2L / v_x) = m v_x² / L.
单个分子对器壁的平均作用力由动量变化率给出：F = Δp / Δt = (2m v_x) / (2L / v_x) = m v_x² / L。
4. Pressure Due to One Molecule | 单个分子产生的压强
Pressure is defined as force per unit area. The area of the wall under consideration is A = L².
压强定义为单位面积上的力。我们所考虑的器壁面积为 A = L²。
The pressure exerted by this single molecule is therefore p = F / A = (m v_x² / L) / L² = m v_x² / L³.
因此，该单个分子产生的压强为 p = F / A = (m v_x² / L) / L² = m v_x² / L³。
Since the volume of the cubic container is V = L³, we can write p = m v_x² / V.
由于立方体容器的体积为 V = L³，我们可以写成 p = m v_x² / V。
5. Considering All N Molecules | 考虑所有 N 个分子
The total pressure P on the wall is the sum of the contributions from all N molecules. Each molecule has its own x-component of velocity.
器壁上的总压强 P 是所有 N 个分子贡献的总和。每个分子都有各自的 x 方向速度分量。
We write the total pressure as P = (Published by TutorHao | IB Physics Revision Series | aleveler.com
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