What is the osmotic pressure (in atm) of a \(1.36 M\) aqueous solution of urea \(\left[\left(\mathrm{NH}_{2}\right)_{2} \mathrm{CO}\right]\) at \(22.0^{\circ} \mathrm{C} ?\)

Chemical thermodynamics involves the study of energy changes accompanying chemical reactions and the study of the relationships between the properties of substances. In the context of osmotic pressure calculation, it plays a pivotal role in understanding how energy is transferred within a solution.

Osmotic pressure itself is a thermodynamic property—it's the pressure that needs to be applied to a solution to prevent the inward flow of water across a semipermeable membrane. This flow of water is driven by differences in free energy—caused by solute concentration differences—and achieving equilibrium is a spontaneous thermodynamic process. When calculating osmotic pressure, we implicitly apply principles of thermodynamics, using equations that relate energy changes to substance concentrations, temperature, and pressure.

Colligative properties are properties of solutions that depend on the number of dissolved particles in the solution, not the type of particles. Osmotic pressure is one of the main colligative properties, along with boiling point elevation, freezing point depression, and vapor pressure lowering.

The reason that the specific identity of the particles doesn't matter for these properties is because they are all linked to how particles affect the solvent at a kinetic or molecular level. For instance, in osmotic pressure, it is the number of solute particles that create an imbalance, driving the flow of solvent through the membrane attempting to equalize the concentration on both sides.

Solution concentration is a measure of the amount of solute that is dissolved in a solvent. It is critical to calculating osmotic pressure, as the principle relies on the solute concentration difference across a membrane. The concentration of a solution can be expressed in various ways, with molarity (M) being one of the most common methods for aqueous solutions.

In our example problem, the osmotic pressure calculation requires the molarity of the urea solution, which is given as 1.36 M. This molarity indicates there are 1.36 moles of urea for every liter of the solution, and this concentration is what drives the osmotic flow of water and thus determines the osmotic pressure.

The gas constant, often denoted by the symbol R, is a crucial constant in the physical sciences used in the ideal gas law and other fundamental equations relating to gases and thermodynamics, like the equation to calculate osmotic pressure. Its value is approximately 0.0821 L.atm/(mol.K) when using atmospheres for pressure, liters for volume, moles for quantity of gas, and Kelvin for temperature.

The gas constant bridges the conceptual gap between the macroscopic and microscopic worlds, correlating the amount of substance and temperature to pressure and volume. In the provided solution, R is used to relate the number of moles of urea, the temperature of the solution, and the osmotic pressure it generates.