The spontaneous mixing of 1000 black beads and 1000 white beads, despite having zero change in enthalpy, can be explained by a positive change in entropy. Entropy measures the number of ways the beads can be arranged (microstates) in the mixed state, which is much higher than when they are segregated. Mixing increases the system's disorder by allowing more possible configurations of the beads, thus increasing entropy. Because the total free energy change (ΔG=ΔH−TΔS\Delta G=\Delta H-T\Delta SΔG=ΔH−TΔS) depends on entropy as well as enthalpy, the increase in entropy drives the spontaneity of the mixing process even when enthalpy change is zero.
Entropy and Mixing
- Initial arrangement: beads are separated into black and white groups with fewer possible configurations.
- Final arrangement: beads are mixed, increasing the number of microstates (ways to redistribute them), leading to higher entropy.
- Entropy of mixing reflects this increase in disorder and possible configurations.
Thermodynamic Implications
- The potential energy (enthalpy) of beads on the left and right is identical, so ΔH=0\Delta H=0ΔH=0.
- Since ΔS>0\Delta S>0ΔS>0 (due to increased randomness), at constant temperature and pressure, the change in Gibbs free energy ΔG=ΔH−TΔS\Delta G=\Delta H-T\Delta SΔG=ΔH−TΔS is negative.
- A negative ΔG\Delta GΔG means the mixing process is spontaneous.
In summary, mixing black and white beads is spontaneous due to the driving force of increased entropy despite no change in enthalpy, because entropy quantifies the increase in disorder and possible arrangements from segregation to mixing.
