Multiple Solid Phases Lecture Notes PDF

Multiple solid phases Many pharmaceutical compounds display multiple crystalline solid state forms Known as crystal polymorphism (e.g., paracetamol: at least three polymorphs) Stability can vary with temperature Can also have amorphous (non-crystalline) solid forms The crystal form o...

Multiple solid phases Many pharmaceutical compounds display multiple crystalline solid state forms Known as crystal polymorphism (e.g., paracetamol: at least three polymorphs) Stability can vary with temperature Can also have amorphous (non-crystalline) solid forms The crystal form of the drug product (medicine) has to be controlled Affects solubility and other key properties Phase diagram for compound with two crystal forms Similar to basic one component phase diagram but two solid regions, S1 and S2 These correspond to two crystal polymorphs Which is preferred (more stable) depends on the conditions (temperature and pressure)  Dealing with non-ideality Defined U, H, S, G with reference to ideal gas systems Need to allow for real (non-ideal) systems which are solids or liquids (esp. solutions) For a system, G = f(P, V, T) Say three components, A, B and C. For an ideal system, total G = GA + GB + GC  Chemical potential For an ideal system, total G = GA + GB + GC Not true for a real system For a real system, any change in composition, temperature or pressure depends on the mutual interactions of all components The contribution of each component to the overall free energy is signified by its chemical potential,   Chemical potential () and free energy (G) Real system at constant  and P Say, three components, A, B and C nA moles of A, nB moles of B, nC moles of C Total free energy of the system = GT,P – (i.e., The free energy at a specific temperature and pressure) 𝐺𝑇,𝑃 = 𝜇𝐴 𝑛𝐴 + 𝜇𝐵 𝑛𝐵 + 𝜇𝐶 𝑛𝐶 A = chemical potential of component A B = chemical potential of component B C = chemical potential of component C  Free energy changes in real systems Recall, for a process at equilibrium: G° = −RTlnKeq System not at equilibrium G − G° = RTlnKeq There is a similar equation for an ideal gas: In this, G is the free energy of the gas, P is the pressure of the gas, G° is its free energy under standard conditions and P° is its pressure under standard conditions. n is the number of moles of the gas. is equivalent to a concentration term in an The term equilibrium constant, i.e., [A]a  Free energy changes in real systems For an ideal gas: The equivalent equation for a real system (e.g., a solution) is: 𝜇𝑖 = 𝜇𝑖° + 𝑅𝑇 ln 𝑎𝑖 In which ai = activity of component i It is activities rather than concentration that should strictly be used in expressions of equilibrium constants Concentrations are used as an approximation that is acceptable depending on the usage  Free energy changes in real systems For a real system, such as a solution: 𝜇𝑖 = 𝜇𝑖° + 𝑅𝑇 ln 𝑎𝑖 Activity (ai) is not the same as concentration ([i]) but is directly proportional to it. The proportionality constant is called the activity coefficient () ai = i[i] i = activity coefficient of component i  Free energy changes in real systems Hence for a process such as The free energy can be given as follows (∆𝐺 ° is the free energy under standard conditions): ∆𝐺 = ∆𝐺 ° + 𝑅𝑇 ln 𝐾 And the equilibrium constant is given by: 𝑎𝐶𝑐 𝑎𝐷𝑑 𝐾= 𝑎 𝑏 𝑎𝐴 𝑎𝐵 (depending on the circumstance, the activities can be replaced by concentrations as an acceptable approximation.)