\rho c_v \frac{\partial T}{\partial t} = \nabla \left ( \kappa T^{n} \nabla T \right )

\rho c_v \frac{\partial T}{\partial t} =\nabla^2 \left ( \frac{1}{n+1} \kappa T^{n+1} \right )

\rho c_v \frac{\partial T}{\partial t} =\nabla^2 \left ( \frac{1}{n+1} \kappa T^{n} T \right )

\rho_i c_v \left ( T_i^{j+1}-T_i^j \right ) = \frac{\kappa \Delta t}{\Delta x^2}\frac{1}{n+1} \left (T_{i+1}^{j^n} T_{i+1}^{j+1} - 2 T_{i}^{j^n} T_{i}^{j+1} + T_{i-1}^{j^n} T_{i-1}^{j+1} \right )

\alpha = \frac{\kappa \Delta t}{\Delta x^2 \left ( n+1 \right ) }

\left (\rho_i c_v + 2 \alpha T_i^{j^n} \right ) T_i^{j+1} - \alpha T_{i+1}^{j^n} T_{i+1}^{j+1} - \alpha T_{i-1}^{j^n} T_{i-1}^{j+1} = \rho_i c_v T_i^j

Now to make the scheme 2nd order in time we can replace T_i^{j+1} with \frac{T_i^{j}+T_i^{j+1}}{2}

\left (\rho_i c_v + 2 \beta T_i^{j^n} \right ) T_i^{j+1} - \beta T_{i+1}^{j^n} T_{i+1}^{j+1} - \beta T_{i-1}^{j^n} T_{i-1}^{j+1} = \rho_i c_v T_i^j - 2 \beta T_i^{j^n} T_i^{j} + \beta T_{i+1}^{j^n} T_{i+1}^{j} + \beta T_{i-1}^{j^n} T_{i-1}^{j}

Which simplifies to:

\left (\rho_i c_v + 2 \beta T_i^{j^n} \right ) T_i^{j+1} - \beta T_{i+1}^{j^n} T_{i+1}^{j+1} - \beta T_{i-1}^{j^n} T_{i-1}^{j+1} = \rho_i c_v T_i^j - 2 \beta T_i^{j^{n+1}} + \beta T_{i+1}^{j^{n+1}} + \beta T_{i-1}^{j^{n+1}}