Consider a colloidal suspension of latex particles confined between two plates. The plates are charged, so there is an electric potential difference V between them (as shown). The number of particles is N and the temperature is T. The system is isolated from the environment and gravitational effects are negligible. Let each particle carries a positive charge +q so that they are more likely to be near the plate at the top in the figure, which is negatively charged, than the bottom one, which is positively charged. The system is very dilute, so particles are independent of each other and the electrostatic interaction between particles can be neglected. For simplicity, assume that in the z direction each particle may only be at one of the following two locations: either attached to the top plate or attached to the bottom one.
(i) Calculate the fraction of particles attached to the top plate. Sketch this as a function of:
(a) T,(b) V. Discuss the physical meaning of your results. Hint/thoughts: Total energy is dependent upon the distribution of the particles between the two plates. The number of configurations associated with a certain energy can be determined, and the entropy of the system can then be expressed as a function of the energy, volume, and number of particles. Using the total energy of the system can be found.
(ii) Consider the case at room temperature (300 K) and q = 10 e, where e is the magnitude of the charge carried by an electron. How large does V need to be so that the particles have a significant preference to reside at the top plate? (q should be regarded as the effective charge carried by a particle).