The excess Gibbs energy of solutions of methylcyclohexane (MCH) and tetrahydrofuran (THF) at 303.15 K was found to tit the expression $$ \mathrm{G}^{\mathrm{E}}=R T x(1-x)\left\\{0.4857-0.1077(2 x-1)+0.0191(2 x-1)^{2}\right\\} $$ where \(\mathrm{x}\) is the mole fraction of the methylcyclohexane. Calculate the Gibbs energy of mixing when a mixture of \(1.00\) mol of MCH and \(3.00\) mol of THF is prepared.

Understanding the excess Gibbs energy is crucial when studying mixtures in physical chemistry. This quantity is a measure of the deviation of a solution's Gibbs energy from the ideal behavior predicted by Raoult's law. An ideal solution would have its components' partial vapor pressures directly proportional to the mole fraction of the respective component.

In real-world scenarios, however, interactions between different molecules can cause this ideal behavior to deviate. Excess Gibbs energy quantifies this deviation and can be either positive or negative, indicating whether the interactions are less or more favorable than the pure components mixing ideally. The formula provided in the exercise, \(\mathrm{G}^{\mathrm{E}}=R T x(1-x)\left\{0.4857-0.1077(2 x-1)+0.0191(2 x-1)^{2}\right\}\), is an empirical expression for excess Gibbs energy in a binary mixture, where \(x\) represents the mole fraction of one component.

Mole fraction is a way of expressing the concentration of a component in a mixture, and it plays a fundamental role in determining mixture properties, such as excess Gibbs energy. It is defined as the ratio of the number of moles of a particular component to the total number of moles of all components in the mixture.

The calculation of mole fraction is straightforward, as seen in the step-by-step solution where \(x = \frac{n_{\mathrm{MCH}}}{n_{\mathrm{MCH}} + n_{\mathrm{THF}}}\). This dimensionless quantity ranges between 0 and 1 and is crucial when using equations that describe the properties of mixtures, such as activity coefficients, vapor pressure, and excess Gibbs energy.

Chemical thermodynamics involves the study of energy changes and transfers that accompany chemical reactions and phase changes. It encompasses the principles governing the direction and extent of chemical processes.

Within thermodynamics, Gibbs energy is a fundamental concept that predicts whether a process will occur spontaneously at constant pressure and temperature. The condition for spontaneity is that the Gibbs energy change must be negative. In terms of mixtures, the Gibbs energy of mixing and the excess Gibbs energy provide insights into the solution behavior and stability. These calculations are essential in fields such as material science, pharmacology, and environmental engineering.

Physical chemistry calculations often involve understanding and predicting the behavior of atoms, molecules, and ions at the microscopic level to explain the macroscopic properties of materials. Calculations like the one in our exercise require accuracy and a thorough knowledge of thermodynamic principles, equations of state, and mathematical skills.

Often, the equations used in such calculations, like the one for excess Gibbs energy, are derived from empirical data or theoretical models. Physical chemistry calculations are not only about plugging numbers into formulas; understanding the implications of these numbers and how they relate to real-world phenomena is what turns these calculations into valuable scientific insight.