Question #34319

1 Answer
Nov 6, 2016

Answer:

#DeltaH > 0#, #DeltaS > 0#

Explanation:

As you know, the only criterion that determines the spontaneity of a reaction is the Gibbs free energy change, #DeltaG#, which is defined as

#color(blue)(bar(ul(|color(black)(DeltaG = DeltaH - T * DeltaS)color(white)(a/a)|)))#

Here

#DeltaH# - the enthalpy change of reaction
#T# - the absolute temperature at which the reaction takes place
#DeltaS# - the entropy change of reaction

Now, in order for a reaction to be spontaneous at a given temperature, it must have

#DeltaG < 0#

This, of course, implies that a non-spontaneous reaction will have

#DeltaG > 0#

A positive Gibbs free energy change corresponds to

#DeltaH - T * DeltaS > 0#

This means that at low temperatures, you have

#DeltaH > T * DeltaS#

Now, this can be true for #DeltaH < 0# and #DeltaS < 0#. However, you are told that at high temperatures the reaction becomes spontaneous.

This means that you need

#DeltaH - T * DeltaS < 0#

or

#DeltaH < T * DeltaS#

As you can see, this cannot be true if #DeltaH < 0# and #DeltaS < 0# because increasing the value of #T# would simply make

#overbrace(DeltaH)^(color(blue)("negative")) > overbrace(T * DeltaS)^(color(blue)("even more negative")) -># non-spontaneous reaction

However, if #DeltaH >0# and #DeltaS > 0#, increasing the value of #T# would make

#overbrace(DeltaH)^(color(darkgreen)("positive")) < overbrace(T * DeltaS)^(color(darkgreen)("even more positive")) -># spontaneous reaction

Remember, #T# is always positive because it expresses absolute temperature.

In general terms, you can have four possible scenarios when dealing with the Gibbs free energy change

  • #DeltaH<0#, #DeltaS>0 -># spontaneous at any temperature
  • #DeltaH>0#, #DeltaS<0 -># non-spontaneous regardless of temperature
  • #DeltaH>0#, #DeltaS>0 -># spontaneous at a certain temperature range
  • #DeltaH<0#, #DeltaS<0 -># spontaneous at a certain temperature range

http://www.eoht.info/page/Gibbs+free+energy

As you can see, reactions that have #DeltaH > 0# and #DeltaS > 0# are only spontaneous at high temperatures.

In this particular case, the reaction is endothermic, since #DeltaH > 0#, but the entropy change of the system overcomes the energy requirement at high temperatures.

A classic example would be the melting of ice, for which

  • #DeltaH > 0 -># you need to add heat to melt ice
  • #DeltaS > 0 -># the entropy of the system is increasing because you're going from solid to liquid

However, the melting of ice is only spontaneous when #T > "273.15 K"#, i.e. at temperatures above #0^@"C"#. When the temperature falls below #0^@"C"#, the melting of ice is a non-spontaneous process.