Dependence of Spontaneity on Temperature:
For a process to be spontaneous , at constant temperature and pressure , there must be decrease in free energy of the system in the direction of the process , i.e. `DeltaG_(P.T) lt 0. DeltaG_(P.T) =0` implies the equilibrium condition and `DeltaG_(P.T) gt 0` corresponds to non- spontaneity.
Gibbs- Helmholtz equation relates the free energy change to the enthalpy and entropy changes of the process as : `" "DeltaG_(P.T) = DeltaH-TDeltaS" ""..."(1)`
The magnitude of `DeltaH` does not change much with the change in temperature but the entropy factor `TDeltaS` change appreciably . Thus, spontaneity of a process depends very much on temperature.
For endothermic process, both `DeltaH` and `DeltaS` are positive . The energy factor, the first factor of equation, opposes the spontaneity whereas entorpy factor favours it. At low temperature the favourable factor `TDeltaS` will be small and may be less than `DeltaH, DeltaG` will have positive value indicated the nonspontaneity of the process. On raising temperature , the factor `TDeltaS` Increases appreciably and when it exceeds `DeltaH, DeltaG` would become negative and the process would be spontaneous .
For an expthermic process, both `DeltaH` and `DeltaS` would be negative . In this case the first factor of eq.1 favours the spontaneity whereas the second factor opposes it. At high temperature , when `T DeltaS gt DeltaH, DeltaG` will have positive value, showing thereby the non-spontaneity fo the process . However , on decreasing temperature , the factor ,`TDeltaS` decreases rapidly and when `TDeltaS lt DeltaH, DeltaG` becomes negative and the process occurs spontaneously. Thus , an exothermic process may be spontaneous at low temperature and non-spontaneous at high temperature.
For the reaction at `298 K ,2A + B rarr C`
`DeltaH =100 kcal` and `DeltaS=0.050 kcal K^(-1)`. If `DeltaH` and `DeltaS` are assumed to be constant over the temperature range, above what temperature will the reaction become spontaneous?

Options

A. 1000 K
B. 1500 K
C. 2000 K
D. 2500 K

Explanation

To determine the temperature above which the reaction becomes spontaneous, we will use the Gibbs-Helmholtz equation:

\[ \Delta G = \Delta H - T \Delta S \]

For the reaction to be spontaneous, we need:

\[ \Delta G < 0 \]

This leads us to the inequality:

\[ \Delta H - T \Delta S < 0 \]

Rearranging this gives:

\[ T \Delta S > \Delta H \]

Thus, we can express the temperature \( T \) as:

\[ T > \frac{\Delta H}{\Delta S} \]

Now, we will substitute the given values into this equation. The values provided are:

- \(\Delta H = 100 \, \text{kcal}\)
- \(\Delta S = 0.050 \, \text{kcal/K}\)

Before substituting, we need to ensure that the units are consistent. Since both values are in kilocalories, we can directly substitute them into the equation.

Calculating \( T \):

\[ T > \frac{100 \, \text{kcal}}{0.050 \, \text{kcal/K}} \]

Calculating the right-hand side:

\[ T > \frac{100}{0.050} = 2000 \, \text{K} \]

Thus, the temperature above which the reaction becomes spontaneous is:

\[ T > 2000 \, \text{K} \]

### Final Answer:
The reaction becomes spontaneous above a temperature of **2000 K**.

---