Question

Consider the following hypothetical equation $$ \mathrm{A}(s)+\mathrm{B}(s) \longrightarrow \mathrm{C}(s)+\mathrm{D}(s) $$ where \(\Delta H^{\circ}=492 \mathrm{~kJ}\) and \(\Delta S^{\circ}=327 \mathrm{~J} / \mathrm{K}\). (a) Obtain an expression for \(\Delta G^{\circ}\) as a function of temperature. Prepare a table of \(\Delta G^{\circ}\) values at \(100 \mathrm{~K}\) intervals $$ \text { between } 100 \mathrm{~K} \text { and } 500 \mathrm{~K} \text { . } $$ (b) Find the temperature at which \(\Delta G^{\circ}\) becomes zero.

Short answer

Question: Calculate the values of ΔG° at temperature intervals of 100 K between 100 K and 500 K, and find the temperature at which ΔG° becomes zero for a given hypothetical reaction with ΔH° = 492 kJ and ΔS° = 327 J/K. Answer: The ΔG° values at every 100 K intervals between 100 K and 500 K are: 460000 J, 428000 J, 396000 J, 364000 J, and 332000 J. The temperature at which ΔG° becomes zero is approximately 1504.6 K.

Step-by-step solution

Recall the relationship between \(\Delta G^{\circ}\), \(\Delta H^{\circ}\), and \(\Delta S^{\circ}\)

The relationship between the standard Gibbs free energy change (\(\Delta G^{\circ}\)), the standard enthalpy change (\(\Delta H^{\circ}\)), and the standard entropy change (\(\Delta S^{\circ}\)) is given by the expression: $$ \Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ} $$ where \(T\) is the temperature in Kelvin.

Calculate \(\Delta G^{\circ}\) as a function of temperature

Since we are given \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\), we can substitute these values into the expression. The resulting equation for \(\Delta G^{\circ}\) as a function of temperature would be: $$ \Delta G^{\circ}(T) = (492 \, \mathrm{kJ}) - T(327 \, \mathrm{J/K}) $$ Don't forget to convert given \(\Delta H^{\circ}\) from kJ to J for consistency: $$ \Delta G^{\circ}(T) = (492\,000 \, \mathrm{J}) - T(327 \, \mathrm{J/K}) $$

Prepare a table for \(\Delta G^{\circ}\) values at 100 K intervals

Now that we have an expression for \(\Delta G^{\circ}\) as a function of temperature, we can calculate its values at temperature intervals of 100 K between 100 K and 500 K. The table for \(\Delta G^{\circ}\) values at each temperature: Temperature (K)|\(\Delta G^{\circ}\) (J) ---|--- 100| 460000 200| 428000 300| 396000 400| 364000 500| 332000

Find the temperature at which \(\Delta G^{\circ}\) becomes zero

To find the temperature at which \(\Delta G^{\circ}\) becomes zero, we simply need to set the expression for \(\Delta G^{\circ}\) equal to zero and solve for \(T\): $$ 492\,000\, \mathrm{J} - T(327 \, \mathrm{J/K}) = 0 $$ Rearrange the equation to solve for \(T\): $$ T = \frac{492\,000\, \mathrm{J}}{327 \, \mathrm{J/K}} $$ Calculate the temperature \(T\): $$ T \approx 1504.6 \mathrm{K} $$ So, the temperature at which \(\Delta G^{\circ}\) becomes zero is approximately 1504.6 K.