Solution Mixing Calculations as a Geometry, Linear Algebra, and Convex Analysis Problem

Abstract

Multicomponent solution calculations can be complicated for students and practiced chemists alike. This article describes how to simplify the calculations by representing a solution’s composition as a point in a “concentration space,” whose axes are the concentrations of each solute. The graphical representation of mixing processes in a concentration space diagram enables qualitative insights such as whether a mixture can be made from a set of stock solutions and the relative volume proportions needed, solely by visual inspection and without detailed calculation. The diagrams correspond to elementary linear algebra equations which facilitates the specification of quantitative calculations. The relationship of the solution mixing problem to convex analysis is discussed. Example applications of this approach to problems of calculating the composition following mixing, determining the volumes to use to create a specified mixture, selecting a set of stock solutions maximizing the number of possible unique mixtures that can be made, random experiment sampling, and common-ion effect problems are demonstrated.

Dimensional Control over Metal Halide Perovskite Crystallization Guided by Active Learning

“…In subsequent cycles of AL, reactions are selected using a diverse-mini-batch-sampling algorithm that prioritizes reaction conditions where model predictions have high uncertainty (5). These reactions are performed using a synthesis robot (6), and the data are used to retrain the model and perform additional cycles of active learning (2)(3)(4)(5)(6). All potential reactions (gray), AL-sampled reactions (colored scattered circles), and prediction uncertainty distribution (colored map) are projected onto 2D space using t-distributed stochastic neighborhood embedding (t-SNE).…”

“…Metal halide perovskite (MHP) derivatives are emerging optoelectronic materials with tunable physical properties [1][2][3][4] and applications in photovoltaic 5,6 and ferroelectric devices, 7 light-emitting diodes, 8,9 and lasers. 9,10 MHP derivatives have been synthesized with diverse crystal structures, with their metal halide frameworks exhibiting connectivities across one, 11 two, 12 or three dimensions.…”

Using automated serendipity to discover how trace water promotes and inhibits lead halide perovskite crystal formation

“…An initial algorithm sampled dispense volumes and was subsequently replaced with an algorithm that performed uniform sampling in the achievable concentration space. 26 The dataset also contains deliberate reproducibility experiments, experimental designs constructed by humans, and experiments guided by machine learning algorithms operating in both exploration and exploitation modes. These different experiment sampling choices may be correlated with seasonal variations in laboratory humidity, and in turn hinder automated serendipity.…”

“…Each batch of 96 experiments consisted of 48 matched pairs, whose compositions were sampled over the possible organoammonium iodide, lead iodide, and formic acid concentration space. 20,26 Sampling strategies included: (a) uniform sampling over all achievable compositions, (b) uniform sampling over compositions where the formic acid concentration was between 7-10 M, (c) uniform sampling within the convex hull defined by previously observed crystal formation.…”

Solution Mixing Calculations for Saturated AC–BC Solutions: A General Analytical Analysis

“…In the paper immediately preceding this one, Schrier 1 has presented an excellent diagrammatic method that leads to the solution of this titration problem. Since his approach is pictorial in nature, it is ideally suited to students who are visual learners.…”