Question

Suppose that a reaction has $\text{ }\Delta\text{ H = 40 kJ}$and$\text{ }\Delta\text{ S = 50 kJ/K}$. At what temperature range it will change from non-spontaneous to spontaneous?
A.$\text{0}\text{.8 K}$to $\text{1}\text{.0 K}$
B.$\text{799 K}$to$\text{800 K}$
C.$\text{800 K}$to $\text{801 K}$
D.$\text{799 K}$to $\text{801 K}$

Answer 1

Hint: The spontaneity of a reaction is decided by the Gibbs free energy of a reaction. If the Gibbs free energy has a negative value then the reaction proceeds with spontaneity, while if the reaction has positive value for Gibbs free energy then the reaction is nonspontaneous.

Complete step by step answer:
The mathematical presentation of the Gibbs free energy \[\text{G = H - TS}\]
The free energy change of a reaction can be defined as, the change in the enthalpy of the reaction $\left( \text{ }\Delta\text{ H} \right)$minus the product of the temperature (Kelvin) and change in the entropy of the system $\left( \text{ }\Delta\text{ S} \right)$
\[\Delta \text{G = }\Delta \text{H - T}\Delta \text{S}\]
Here, $\text{ }\Delta\text{ H = 40 kJ}$ and $\text{ }\Delta\text{ S = 50 kJ/K}$
Now as said earlier,
If \[\Delta \text{G }\]< 0, the process is a spontaneous one,
If \[\Delta \text{G }\]> 0, the process is a non-spontaneous one,
\[\Delta \text{G }\]= 0, the process is at equilibrium.
\[\Delta \text{G = }40\text{ - T}\times 50\]
If the temperature is$\text{0}\text{.8 K}$, \[\Delta \text{G = }40\text{ - }\left( 0.8\times 50 \right)=0\]the system is at equilibrium, while at $\text{1}\text{.0 K}$, \[\Delta \text{G = }40\text{ - }\left( 1\times 50 \right)=-10\]kJ, the process is spontaneous.
If the temperature is$\text{799 K}$, \[\Delta \text{G = }40\text{ - }\left( 799\times 50 \right)=-39,910\]the system is at equilibrium, while at $\text{800}\text{.0 K}$, \[\Delta \text{G = }40\text{ - }\left( 800\times 50 \right)=-39,960\]kJ, the process is spontaneous.
If the temperature is $\text{799 K}$, \[\Delta \text{G = }40\text{ - }\left( 799\times 50 \right)=-39,910\]the process is spontaneous, while at $\text{800}\text{.0 K}$, \[\Delta \text{G = }40\text{ - }\left( 800\times 50 \right)=-39,960\]kJ, the process is spontaneous.
If the temperature is $\text{801 K}$, \[\Delta \text{G = }40\text{ - }\left( 801\times 50 \right)=-40,010\], the process is spontaneous.
Among all the processes,

The one in which the process will become spontaneous from a non-spontaneous one is option A, $\text{0}\text{.8 K}$to $\text{1}\text{.0 K}$, which is the correct Option.
Note:
The Gibbs free energy is defined as the energy change that is associated with a chemical change that is used to do work. The free energy of a system is the sum of the enthalpy of the system (H) and the product of the temperature (Kelvin) and the entropy of the system (S).