A process will be spontaneous at all temperatures if:

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Strategy:
 A process is spontaneous if . To be spontaneous at all temperatures (), the relationship must yield a negative  regardless of the value of  (where ).

Step 1: Evaluate the conditions
- If  (exothermic) and  (increasing entropy):
  - 
  - This expression will always be negative because both terms are negative.
- If  and :  is always positive (non-spontaneous).
- For other combinations, spontaneity depends on the specific magnitude of .

Step 2: Conclusion
The condition  and  ensures sp

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