Step 1: Recall the definition of an ideal solution.
An ideal solution is a solution that obeys Raoult's law over the entire range of concentration. In such solutions, the intermolecular forces between unlike molecules are nearly the same as those between like molecules. Because of this, mixing occurs without any significant heat change or volume change.
Step 2: Enthalpy change of mixing.
For an ideal solution, the energy required to separate molecules of the pure components is almost exactly equal to the energy released when unlike molecules interact after mixing. Therefore, there is no net heat absorbed or evolved. Hence,
\[
\Delta_{\mathrm{mix}} H = 0
\]
Step 3: Volume change of mixing.
In an ideal solution, since the sizes and interactions of the molecules are similar, the total volume after mixing remains equal to the sum of the individual volumes before mixing. Thus, there is no contraction or expansion in volume. Hence,
\[
\Delta_{\mathrm{mix}} V = 0
\]
Step 4: Conclusion.
Therefore, the correct condition for an ideal solution is
\[
\Delta_{\mathrm{mix}} H = 0 \quad \text{and} \quad \Delta_{\mathrm{mix}} V = 0
\]
Final Answer:\( \Delta_{\mathrm{mix}} H = 0;\ \Delta_{\mathrm{mix}} V = 0 \).