1. Interpret the given thermochemical equations

   The reactions are written as heat being evolved:

   $\text{C(diamond)} \rightarrow \text{C(graphite)} + X\ \text{kJ mol}^{-1}$ $\text{C(diamond)} + O_2(g) \rightarrow CO_2(g) + Y\ \text{kJ mol}^{-1}$ $\text{C(graphite)} + O_2(g) \rightarrow CO_2(g) + Z\ \text{kJ mol}^{-1}$

   Since heat appears on the product side, the enthalpy changes are:

   $\Delta H_1 = -X$ $\Delta H_2 = -Y$ $\Delta H_3 = -Z$

2. Apply Hess's law

   To go from diamond to $CO_2$, we can follow two paths:

   • Direct path: $\text{C(diamond)} + O_2 \rightarrow CO_2$ with enthalpy change $-Y$.

   • Indirect path: first convert diamond to graphite, then burn graphite: $\text{C(diamond)} \rightarrow \text{C(graphite)} \qquad \Delta H=-X$ $\text{C(graphite)} + O_2 \rightarrow CO_2 \qquad \Delta H=-Z$

     Total enthalpy change: $(-X) + (-Z) = -(X+Z)$

   By Hess's law, both paths must have the same enthalpy change:

   $-Y = -X - Z$

3. Rearrange

   $Y = X + Z$

   Hence,

   $X = Y - Z$

4. Check options

   • A: $-X = Y + Z$ ❌
   • B: $X = Y - Z$ ✅
   • C: $X = -Y + Z$ ❌
   • D: $X = Y + Z$ ❌

Therefore, the correct option is B.