Edexcel Mathematics 2.5: Biological Membranes | 爱德思数学 2.5 生物膜

📚 Edexcel Mathematics 2.5: Biological Membranes | 爱德思数学 2.5 生物膜

In Edexcel Mathematics, contextual problems often draw on biological systems such as cell membranes. This article explains how core mathematical techniques can model diffusion, osmosis, surface area effects and membrane potentials.

在爱德思数学中，情境题常以生物系统（如细胞膜）为背景。本文解释如何用核心数学技巧对扩散、渗透、表面积效应和膜电位进行建模。

1. Surface Area to Volume Ratio | 表面积与体积比

For a cube-shaped cell with side length x, surface area is 6x² and volume is x³. The ratio SA:V = 6/x, so as x increases the ratio decreases.

对于边长为 x 的立方体细胞，表面积为 6x²，体积为 x³。表面积体积比 SA:V = 6/x，因此当 x 增大时比值减小。

SA : V = 6x² : x³ = 6 : x

Small cells and folded membranes both increase the effective SA:V ratio. In exam questions, you may be asked to compare two cubes or spheres by calculating percentage change.

小细胞和折叠膜都能增大有效的表面积体积比。考试题中可能要求你通过计算百分比变化来比较两个立方体或球体。

For a sphere, SA = 4πr² and V = (4/3)πr³, giving SA:V = 3/r. This is another standard result to memorise.

对于球体，表面积 SA = 4πr²，体积 V = (4/3)πr³，因此 SA:V = 3/r。这是另一个需要记住的标准结果。

2. Fick’s Law and Proportional Reasoning | 菲克定律与比例推理

Fick’s law states that diffusion rate is directly proportional to surface area and concentration gradient, and inversely proportional to membrane thickness. Mathematically, Rate = D × A × ΔC ÷ d.

菲克定律指出，扩散速率与表面积和浓度梯度成正比，与膜厚度成反比。数学表达式为 Rate = D × A × ΔC ÷ d。

Rate = D × A × ΔC ÷ d

Here D is the diffusion coefficient. Questions often give three variables and ask you to find the fourth, or predict the effect of doubling membrane thickness.

其中 D 是扩散系数。题目通常给出三个变量并要求求第四个，或预测膜厚度加倍对速率的影响。

Remember that doubling the concentration gradient doubles the rate, while doubling the thickness halves the rate. This is direct and inverse proportion in action.

记住：浓度梯度加倍会使速率加倍，而厚度加倍会使速率减半。这正是正比例和反比例的应用。

3. Osmotic Pressure and the van ‘t Hoff Equation | 渗透压与范特霍夫方程

Osmotic pressure π is given by π = iCRT, where i is the van ‘t Hoff factor, C is molar concentration, R is the gas constant and T is absolute temperature.

渗透压 π 由公式 π = iCRT 给出，其中 i 是范特霍夫因子，C 是摩尔浓度，R 是气体常数，T 是绝对温度。

π = i × C × R × T

The equation is a linear model if i, R and T are constant. Edexcel papers may ask you to rearrange for C or to compare osmotic pressures under different concentrations.

如果 i、R 和 T 为常数，该方程

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