1. Given:

   • Monoatomic gas: $2$ moles
   • Diatomic gas: $6$ moles

2. Molar specific heat at constant volume for each gas:

   • For a monoatomic ideal gas: $C_{V1} = \frac{3}{2}R$
   • For a diatomic ideal gas: $C_{V2} = \frac{5}{2}R$

3. Total heat capacity at constant volume of the mixture: $C_V^{\text{total}} = n_1 C_{V1} + n_2 C_{V2}$ $= 2\left(\frac{3}{2}R\right) + 6\left(\frac{5}{2}R\right)$ $= 3R + 15R = 18R$

4. Total number of moles in the mixture: $n_{\text{total}} = 2 + 6 = 8$

5. Molar specific heat of the mixture at constant volume: $C_{V,m} = \frac{C_V^{\text{total}}}{n_{\text{total}}}$ $= \frac{18R}{8} = \frac{9}{4}R$

6. Therefore, the correct option is: $\boxed{\frac{9}{4}R}$

So, Option D is correct.