Correct relationship between heat of fusion `(DeltaH_(fus))`, heat of vaporization `(DeltaH_("vap"))` and heat of sublimation `(DeltaH_("sub"))` is

Options

A. `DeltaH_("fus")=DeltaH_("vap")+DeltaH_("sub")`
B. `DeltaH_("vap")=DeltaH_("fus")+DeltaH_("sub")`
C. `DeltaH_("sub")=DeltaH_("vap")+DeltaH_("fus")`
D. `DeltaH_("sub")=DeltaH_("vap")-DeltaH_("fus")`

Answer

To find the correct relationship between the heat of fusion (ΔH_fus), heat of vaporization (ΔH_vap), and heat of sublimation (ΔH_sub), we can analyze the processes involved in each term.

### Step-by-Step Solution:

1. **Define the Terms**:
- **Heat of Fusion (ΔH_fus)**: This is the amount of energy required to convert a solid into a liquid at its melting point without changing its temperature.
- **Heat of Vaporization (ΔH_vap)**: This is the amount of energy required to convert a liquid into a gas at its boiling point without changing its temperature.
- **Heat of Sublimation (ΔH_sub)**: This is the amount of energy required to convert a solid directly into a gas without passing through the liquid phase.

2. **Write the Processes**:
- For heat of fusion:
\[ \text{Solid} \xrightarrow{\Delta H_{fus}} \text{Liquid} \]
- For heat of vaporization:
\[ \text{Liquid} \xrightarrow{\Delta H_{vap}} \text{Gas} \]
- For heat of sublimation:
\[ \text{Solid} \xrightarrow{\Delta H_{sub}} \text{Gas} \]

3. **Combine the Processes**:
- If we combine the first two processes (solid to liquid and then liquid to gas), we can express this as:
\[ \text{Solid} \xrightarrow{\Delta H_{fus}} \text{Liquid} \xrightarrow{\Delta H_{vap}} \text{Gas} \]
- The total energy required for this combined process is:
\[ \Delta H_{fus} + \Delta H_{vap} \]

4. **Set Up the Equation**:
- Since the direct conversion from solid to gas (sublimation) requires the same total energy, we can equate the two expressions:
\[ \Delta H_{fus} + \Delta H_{vap} = \Delta H_{sub} \]

5. **Conclusion**:
- The correct relationship is:
\[ \Delta H_{sub} = \Delta H_{fus} + \Delta H_{vap} \]

### Final Answer:
The correct relationship is:
\[ \Delta H_{sub} = \Delta H_{fus} + \Delta H_{vap} \]