Consider a circuit When the switch S is opened C1 is charged to voltage Vo and C2 is uncharged.  After the switch S is closed, the amount of stored energy lost due to heat is

$\frac{1}{2} C_0V_0^2$ $\frac{1}{4} C_0V_0^2$ $\frac{3}{4} C_0V_0^2$ $\frac{1}{8} C_0V_0^2$

$\frac{1}{4} C_0V_0^2$

Loss of energy due to heat is $\Delta U = \frac{C_1C_2}{2(C_1+C_2)} (V_1-V_2)^2$  $\Delta U = \frac{C_0C_0}{2(C_0+C_0)} (V_0-0)^2 = \frac{1}{4} C_0V_0^2$