基于GeoGebra的交互式色谱初学者学习工具开发与实践——以塔板理论为例

doi:10.3724/SP.J.1123.2025.03008

摘要/Abstract

摘要：

针对仪器分析课程色谱教学中存在的理论表达抽象、知识传授单向、学生参与度低等问题，本文以塔板理论为切入点，构建了基于GeoGebra平台的交互式色谱学习工具。针对现有教学工具存在专业软件操作复杂、网页工具访问受限、参数调控自由度不足等缺陷，提出了“模型重构-工具开发-教学链融合”的创新型解决方案。通过整合流动相流速、死时间和相比等关键参数，建立改进型塔板理论数学模型；基于云端计算平台开发包含单组分模拟、多组分模拟和保留时间方程推导的三级进阶式学习模块；创新设计覆盖“数字化建模（‘豆包’AI辅助推导）-参数交互（多项可调色谱参数）-可视化验证（色谱流出曲线模拟）”的完整教学链。教学实践表明：（1）该工具突破了传统理论教学维度限制，课堂任务完成率提升至94%，学生高阶问题解答正确率提高至76%；（2）动态参数调控功能显著增强了学习参与度，85%的学生在后续学习中可自主使用该工具；（3）AI辅助推导与回归分析模块实现了理论化学与计算工具的跨学科融合，通过推导得出保留时间方程的做法比目前教材中直接给出结论的方式更具说服力。研究表明，这种“理论模型可视化-模型参数可调化-知识生成交互化”的创新模式，为解决色谱理论教学困境提供了新路径，其开源框架和模块化设计理念可为分析化学数字化教学改革提供有益参考。

关键词: 仪器分析课程, 塔板理论, GeoGebra 平台, 交互式学习, 色谱流出曲线模拟, 数字化教学

Abstract:

This study developed a GeoGebra platform-based interactive pedagogical tool focusing on plate theory to address challenges associated with abstract theory transmission， unidirectional knowledge delivery， and low student engagement in chromatography teaching in instrumental analysis courses. This study introduced an innovative methodology that encompasses theoretical model reconstruction， tool development， and teaching-chain integration that addresses the limitations of existing teaching tools， including the complex operation of professional software， restricted accessibility to web-based tools， and insufficient parameter-adjustment flexibility. An improved mathematical plate-theory model was established by incorporating mobile-phase flow rate， dead time， and phase ratio parameters. A three-tier progressive learning system （single-component simulation， multi-component simulation， and retention-time-equation derivation modules） was developed on a cloud-based computing platform. An integrated teaching chain that combined athematical modeling （AI-assisted “Doubao” derivation）， interactive-parameter adjustment （multiple adjustable chromatographic parameters）， and visual verification （chromatographic elution-curve simulation） was implemented. Teaching practice demonstrated that：（1） The developed tool transcends the dimensional limitations of traditional instruction， elevating the classroom task completion rate to 94% and improving the student accuracy rate for solving advanced problems to 76%. （2） The dynamic-parameter-adjustment feature significantly enhances learning engagement by enabling 85% of the students to independently use the tool in subsequent studies and experiments. （3） The AI-powered derivation and regression-analysis modules enable the interdisciplinary integration of theoretical chemistry and computational tools. The process of deriving chromatographic retention-time equations through this methodological approach proved more convincing than the current textbook practice of directly presenting conclusions. The developed innovative “theoretical-model visualizable-model-parameter adjustable-interactive-knowledge generating” model provides a new avenue for addressing teaching challenges associated with chromatography theory， and its open-source framework and modular design philosophy can offer valuable references for the digital teaching reform in analytical chemistry.

Key words: instrumental analysis course, plate theory, GeoGebra platform, interactive learning, chromatographic elution profile simulation, digital teaching

中图分类号:

O658

张雨晗, 何珺瑶, 林淑婧, 艾炳健, 石志红, 张红医. 基于GeoGebra的交互式色谱初学者学习工具开发与实践——以塔板理论为例[J]. 色谱, 2025, 43(9): 1078-1085.

ZHANG Yuhan, HE Junyao, LIN Shujing, AI Bingjian, SHI Zhihong, ZHANG Hongyi. Development and practice of an interactive chromatography learning tool for beginners based on GeoGebra： a case study of plate theory[J]. Chinese Journal of Chromatography, 2025, 43(9): 1078-1085.

图/表 9

表1 用GeoGebra模拟单组分样品色谱流出曲线的流程

Table 1 Procedures of operation for simulated chromatographic elution profile of single-component sample by GeoGebra

Line No.	Input	Comment
1	$C 0$ =1	initial concentration
2	$t m = 3$	dead time
3	$β = 2$	phase ratio
4	$n = 100$	number of plates
5	$K = 3$	distribution coefficient
6	$v (t) = t n (K / β + 1) t m$	Equation （9）^*
7	$f (t) = 1 1 + K C 0 1 2 π n (v (t) n) n ⋅ e n - v (t)$	Equation （4）^*

表1 用GeoGebra模拟单组分样品色谱流出曲线的流程

Table 1 Procedures of operation for simulated chromatographic elution profile of single-component sample by GeoGebra

Line No.	Input	Comment
1	$C 0$ =1	initial concentration
2	$t m = 3$	dead time
3	$β = 2$	phase ratio
4	$n = 100$	number of plates
5	$K = 3$	distribution coefficient
6	$v (t) = t n (K / β + 1) t m$	Equation （9）^*
7	$f (t) = 1 1 + K C 0 1 2 π n (v (t) n) n ⋅ e n - v (t)$	Equation （4）^*

图1 GeoGebra平台上单一组分的色谱流出曲线模拟图

Fig. 1 Simulated chromatographic elution profile of single-component sample by GeoGebra

表2 用GeoGebra模拟双组分样品（K=3和1.8）色谱流出曲线的流程

Table 2 Procedures of operation for simulated chromatographic elution profiles of dual-component sample （K=3 and 1.8） by GeoGebra

Line No.	Input	Comment
1	$C 0$ = 1	initial concentration of component A
2	$t m = 3$	dead time
3	$β = 2$	phase ratio
4	$n = 100$	number of plates
5	$v (t, K) = t n t m (1 + K / β)$	Equation （9）^*
6	$f (t, K) = 1 1 + K C 0 1 2 π n (v (t, K) n) n ⋅ e n - v (t, K)$	Equation （4）^*
7	$f (t, 3)$	simulating profile of $K = 3$
8	$f (t, 1.8)$	simulating profile of $K = 1.8$

表2 用GeoGebra模拟双组分样品（K=3和1.8）色谱流出曲线的流程

Table 2 Procedures of operation for simulated chromatographic elution profiles of dual-component sample （K=3 and 1.8） by GeoGebra

Line No.	Input	Comment
1	$C 0$ = 1	initial concentration of component A
2	$t m = 3$	dead time
3	$β = 2$	phase ratio
4	$n = 100$	number of plates
5	$v (t, K) = t n t m (1 + K / β)$	Equation （9）^*
6	$f (t, K) = 1 1 + K C 0 1 2 π n (v (t, K) n) n ⋅ e n - v (t, K)$	Equation （4）^*
7	$f (t, 3)$	simulating profile of $K = 3$
8	$f (t, 1.8)$	simulating profile of $K = 1.8$

图2 GeoGebra平台上两组分的色谱流出曲线模拟图

Fig. 2 Simulated chromatographic elution profiles of dual-component sample by GeoGebra

表3 用GeoGebra模拟不同分配系数（K=1，3，5，7，9）样品色谱流出曲线的流程

Table 3 Procedures of operation for simulated chromatographic elution profiles of five-component sample （K=1，3，5，7，9） by GeoGebra

Line No.	Input	Comment
1	$C 0$ = 1	initial concentration of the sample
2	$t m = 3$	dead time
3	$β = 2$	phase ratio
4	$n = 100$	number of plates
5	$v (t, K) = t n t m (1 + K / β)$	Equation （9）^*
6	$f (t, K) = 1 1 + K C 0 1 2 π n (v (t, K) n) n ⋅ e n - v (t, K)$	Equation （4）^*
7	$f (t, 1)$	simulating profile of $K = 1$
8	$f (t, 3)$	simulating profile of $K = 3$
9	$f (t, 5)$	simulating profile of $K = 5$
10	$f (t, 7)$	simulating profile of $K = 7$
11	$f (t, 9)$	simulating profile of $K = 9$

表3 用GeoGebra模拟不同分配系数（K=1，3，5，7，9）样品色谱流出曲线的流程

Table 3 Procedures of operation for simulated chromatographic elution profiles of five-component sample （K=1，3，5，7，9） by GeoGebra

Line No.	Input	Comment
1	$C 0$ = 1	initial concentration of the sample
2	$t m = 3$	dead time
3	$β = 2$	phase ratio
4	$n = 100$	number of plates
5	$v (t, K) = t n t m (1 + K / β)$	Equation （9）^*
6	$f (t, K) = 1 1 + K C 0 1 2 π n (v (t, K) n) n ⋅ e n - v (t, K)$	Equation （4）^*
7	$f (t, 1)$	simulating profile of $K = 1$
8	$f (t, 3)$	simulating profile of $K = 3$
9	$f (t, 5)$	simulating profile of $K = 5$
10	$f (t, 7)$	simulating profile of $K = 7$
11	$f (t, 9)$	simulating profile of $K = 9$

图3 GeoGebra平台上5组分的色谱流出曲线模拟图

Fig. 3 Simulated chromatographic elution profile of five-component sample by GeoGebra

表4 用GeoGebra进行保留时间和分配系数线性回归分析的步骤

Table 4 Operational procedures for linear regression analysis between retention time and distribution coefficient with GeoGebra

Line No.	Input	Comment
1	A=（1，4.5）	$K = 1, t = 4.5$
2	B=（3，5，3）	$K = 3, t = 7.5$
3	C=（5，10.5）	$K = 5, t = 10.5$
4	D=（7，13.5）	$K = 7, t = 13.5$
5	E=（9，16.5）	$K = 9, t = 16.5$
6	fitline （｛A，B，C，D，E｝）	conducting linear fitting
7	corresponding coefficient （｛A，B，C，D，E｝）	showing correlation coefficient

表4 用GeoGebra进行保留时间和分配系数线性回归分析的步骤

Table 4 Operational procedures for linear regression analysis between retention time and distribution coefficient with GeoGebra

Line No.	Input	Comment
1	A=（1，4.5）	$K = 1, t = 4.5$
2	B=（3，5，3）	$K = 3, t = 7.5$
3	C=（5，10.5）	$K = 5, t = 10.5$
4	D=（7，13.5）	$K = 7, t = 13.5$
5	E=（9，16.5）	$K = 9, t = 16.5$
6	fitline （｛A，B，C，D，E｝）	conducting linear fitting
7	corresponding coefficient （｛A，B，C，D，E｝）	showing correlation coefficient

图4 GeoGebra计算云平台的（a）界面和（b）工作界面

Fig. 4 （a） Interface of GeoGebra calculation cloud platform and （b） its working interface

图5 两组分色谱峰模拟的主要键入内容和流出曲线模拟结果界面

Fig. 5 Main input contents and simulated chromatographic elution profiles of dual-component sample by GeoGebra

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