1. 扩散第二定律的导出                                                                 
    在垂直于扩散方向x的方向上取一截面积为A、长度为dx的体积元。设流入和流出体积元的通量分别为J₁和J₂，则在dt时间内                                                  
流入量为：                                                                         
流出量为：                                              
浓度增加量为： 即                             
所以有：                                            
将第一定律带入，得 ——扩散第二定律                             
设D与c无关，则上式可简化为                                     
    上式表明：浓度随时间的变化率与浓度分布曲线的二阶导数成正比。                                 
                                                                                
2. 第一定律和第二定律对扩散过程的描述                                           
    对扩散过程的描述一致：扩散过程使不均匀体系均匀化，有由非平衡逐渐达到平衡，如图所示。                                                                            

                                                                  
                                                      
                                                 
                            
    扩散第二定律是一个偏微分方程，采用不同的解法可得到适用于不同情况的解。常见的解有误差函数解、高斯解和正弦解。                                                      
3. 误差函数解                                                                        
扩散第二定律的误差解为                                                           
                                    
    上式即扩散第一定律的误差解的通解，其中为误差函数，其值可以通过查表求得。误差函数具有如下性质                                          
                                                                                 
                                                                                   
                                                                                    
    误差解（通解）－适用于扩散物质开始均匀分布在一个大范围的扩散问题，如金属的氧化、脱碳及化学热处理等扩散问题均可用误差解处理。                                       
下面以钢的渗碳为例说明误差解的应用。                                               
    假定渗碳开始后表面奥氏体含碳量立即达到c_S并保持恒定，工件心部始终为其原始成分c₀，则                                                                              
初始条件：                                                        
边界条件：                                                                           
                                                                                       
将边界条件带入误差解，并利用误差函数的性质，则得                                      
                                                                                       
                                                                                     
                                                                                      
                                                                                       
                                                                                   
上式描述了表层碳浓度与位置和时间的关系，并可进一步确定深层深度与时间的关系:          
若规定某一浓度c_m 对应的深度x为渗层深度，则                                              
，即常数，所以                                           
                                                                              
式中k为常数。上式表明，渗层深度与时间成抛物线关系。