Enthalpy of Sublimation, Hsubl

The heat (or enthalpy) of sublimation is the amount of energy that must be added to a solid at constant pressure in order to turn it directly into a gas (without passing through the liquid phase).  The heat of sublimation is generally expressed as ΔH_sub in units of Joules or kiloJoules per mole or kilogram of substance.

1. 1. Things to recall...
2. 2. Introduction to Sublimation and Heat of Sublimation
3. 3. The Equations
   1. 3.1. ΔHsub = ΔEtot
      1. 3.1.1. ΔEthermal(state of matter)
      2. 3.1.2. ΔEbond(going from state 1 to state 2)
4. 4. Graphical Representations of the Heat of Sublimation
5. 5. ΔHsub > ΔHvap
6. 6. Where does the added energy go?
7. 7. Practical Applications of the Heat of Sublimation
8. 8. Practice Problems
9. 9. Practice Problem Solutions
10. 10. Footnotes
11. 11. External References
12. 12. Contributors

Things to recall...

1. A "Δ" in front of any variable indicates that the following equation is a state function in which the value of an indicator measured in the initial state of a reaction is subtracted from the value of the same indicator measured in the final state of the reaction.  The resulting value is the change in the indicator at the end of the said reaction (Δindicator). 
2. Enthalpy is the amount of energy that is required to induce a phase change (change in the state of matter).
3. Energy is measured in Joules or kiloJoules and is transferred through either through heat (q) or work (w).
4. The phases involved in sublimation are the solid phase and the gas phase.

Introduction to Sublimation and Heat of Sublimation

Sublimation is the process of changing a solid into a gas without allowing the solid to pass through the liquid phase.  In order to sublime a substance, a certain amount of energy must be transferred to the substance via heat (q) or work (w).  The energy needed to sublime a substance is particular to the substance's identity and temperature and must be enough to do all of the following:

1. Excite the substance_(s) so that it reaches its maximum heat (energy) capacity (q) in the solid state.
2. Sever all the atomic bonds in the substance_(s).
3. Excite the unbonded atoms of the substance so that it reaches its minimum heat capacity in the gaseous state.

(See attached powerpoint show in the Files section at the bottom of the page.  Select the option to play slideshow.)

The Equations

ΔH_{sub =} ΔE_tot

The energies involved in sublimation steps 1 through 3 (see Sublimation ) compose the total amount of energy that is involved in sublimation.  This can be expressed by the equation,

in which

Although a solid does not actually pass through the liquid phase during the process of sublimation, the fact that enthalpy is a state function allows us to add the various energies associated with the solid, liquid, and gas phases together.  Recall that for state functions, only the initial and final states of the substance are important.  Say for example that state A is the initial state and state B is the final state.  How a substance goes from state A to state B does not matter so much as what state A and what state B are.  Concerning the state function of enthalpy, the energies associated with enthalpies (whose associated states of matter are contiguous to one another) are additive.  Though in sublimation a solid does not pass through the liquid phase on its way to the gas phase, it takes the same amount of energy that it would to first melt (fuse) and then vaporize. 

ΔE_{thermal(state of matter)}

A change in thermal energy is indicated by a change in temperature (in Kelvin) of a substance at any particular state of matter.  Change in thermal energy is expressed by the equation

in which

For more information on heat capacity and specific heat capacity, see heat capacity.  Specific heat capacities of common substances can easily be found online or in a reference or text book.

ΔE_{bond(going from state 1 to state 2)}

Bond energy is the amount of energy that a group of atoms must absorb so that it can undergo a phase change (going from a state of lower energy to a state of higher energy).  It is measured

in which is the enthalpy associated with a specific substance at a specific phase change.  Common types of enthalpies include the heat of fusion (melting) and the heat of vaporization.  Recall that fusion is the phase change that occurs between the solid state and the liquid state, and vaporization is the phase change that occurs between the liquid state and the gas state.  Note that if the substance has more than one type of bond (or intramolecular force), then the substance must absorb enough energy to break all the different types of bonds before the substance can sublime.¹

Graphical Representations of the Heat of Sublimation

Note that although the graph below indicates the inclusion of the liquid phase, the graph is merely a representation of how much energy is needed to sublime a solid substance.  Recall that sublimation does not include the liquid phase and that the fact that enthalpy is a state function allows us to add the enthalpies of fusion and vaporization together to find the enthalpy of sublimation.

        

ΔH_{sub >} ΔH_vap

Though both enthalpies involve the changing a substance into its gaseous state, the change in energy associated with sublimation is generally greater than that of vaporization.  This is because of the initial state of the substances and the amount of initial energy that each substance has.  Particles in a solid have less energy than those of a liquid, meaning it is takes more energy to excite a solid to its gaseous phase that it does to excite a liquid to its gaseous phase. 

Another way to look this phenomena is to take a look at the different energies involved with the heat of sublimation:  ΔE_{thermal (s)}, ΔE_bond(s-l), ΔE_thermal(l), and ΔE_bond(l-g).  Already we know that ΔE_bond=ΔH_{(phase change)}*Δm_{(changed substance)} and ΔE_bond(l-g)=ΔH_(l-g)*Δm_{(gas created)}.  ΔH_(l-g) is essentially ΔH_{(vaporization)}, meaning that ΔH_vap is actually a component of ΔH_sub.

 

Where does the added energy go?

Energy can be observed in many different ways.  As shown above, ΔE_tot can be expressed as ΔE_{thermal +} ΔE_bond.  Another way in which  ΔE_tot can be expressed is change in potential energy, ΔPE, plus change in kinetic energy, ΔKE.  Potential energy is the energy associated with random movement, whereas kinetic energy is the energy associated with velocity (movement with direction).  ΔE_tot = ΔE_{thermal +} ΔE_bond and ΔE_tot = ΔPE + ΔKE are related by the equations

ΔPE = (0.5)ΔE_{thermal +} ΔE_bond ΔKE = (0.5)ΔE_thermal

for substances in the solid and liquid states.  Note that ΔE_thermal is divided equally between ΔPE and ΔKE for substances in the solid and liquid states.  This is because the intermolecular and intramolecular forces that exist between the atoms of the substance (i.e. atomic bond, van der Waals forces, etc) have not yet been dissociated and prevent the atomic particles from moving freely about the atmosphere (with velocity).  Potential energy is just a way to have energy, and it generally describes the random movement that occurs when atoms are forced to be close to one another.  Likewise, kinetic energy is just another way to have energy, which describes an atom's vigorous struggle to move and to break away from the group of atoms.  The thermal energy that is added to the substance is thus divided equally between the potential and the kinetic energies  because all aspects of the atoms' movement must be excited equally

However, once the intermolecular and intramolecular forces which restrict the atoms' movement are dissociated (when enough energy has been added), potential energy no longer exists (for monatomic gases) because the atoms of the substance are no longer forced to vibrate and be in contact with other atoms.  When a group of atoms is in the gaseous state, it's atoms can devote all their energies into moving away from one another (kinetic energy).