Solutions of hydrogen in palladium may be formed by exposing Pd metal to \(\mathrm{H}_{2}\) gas. The concentration of hydrogen in the palladium depends on the pressure of \(\mathrm{H}_{2}\) gas applied, but in a more complex fashion than can be described by Henry's law. Under certain conditions, 0.94 g of hydrogen gas is dissolved in 215 g of palladium metal (solution density \(=10.8 \mathrm{g} \mathrm{cm}^{3}\) ). (a) Determine the molarity of this solution. (b) Determine the molality of this solution. (c) Determine the percent by mass of hydrogen atoms in this solution.

Understanding molarity is crucial when it comes to solution concentration calculations. It measures the concentration of a solution by indicating how many moles of solute are present in one liter of solution. You can consider molarity as a way to quantify the 'strength' of a solution in terms of how much of the substance of interest is contained within a specific volume of liquid.

The formula to calculate molarity (M) is as follows: \[ M = \frac{\text{moles of solute}}{\text{volume of solution in liters}} \]
For instance, if you have calculated the number of moles of hydrogen in a solution and you know the volume of that solution in liters, you can find the molarity by simply dividing the moles by the volume. This is frequently used in chemistry to prepare solutions with precise concentrations and to perform stoichiometric calculations in reaction equations.

Molality provides another measure of solution concentration and is especially useful when dealing with temperature changes, as it isn't affected by variations in temperature. While molarity is dependant on the solution's volume, which can expand or contract with temperature, molality relies on the mass of the solvent.

The formula for molality (m) is expressed as: \[ m = \frac{\text{moles of solute}}{\text{mass of solvent in kilograms}} \]
It's simply the number of moles of the solute divided by the kilograms of the solvent. This distinction from molarity makes molality important for colligative properties studies, such as boiling point elevation and freezing point depression, where the mass of the solvent remains constant regardless of ambient conditions.

The percent by mass (also known as weight percent or mass percent) is a concentration measure that communicates the mass of the solute relative to the total mass of the solution. It's a ratio that tells you how many grams of solute are in 100 grams of solution, thus providing an easy way to understand the composition of mixtures in everyday terms.

The formula to determine the percent by mass is: \[ \text{Percent by mass} = \left( \frac{\text{mass of solute}}{\text{mass of solute} + \text{mass of solvent}} \right) \times 100\% \]
This measure is valuable for solutions where the solute and solvent mass can be easily distinguished and is commonly used in solid and semi-solid product formulations, such as pharmaceuticals or food products.

When dealing with specific solutions such as hydrogen in palladium, it is important to consider that the solubility might not follow a simple law such as Henry's law, which is often the case for gases that dissolve in metals. The interaction of hydrogen with palladium causes the metal to absorb the hydrogen, creating a 'solution' that does not behave like typical liquid solutions.

In any case, traditional measures of concentration such as molarity, molality, and percent by mass can still be applied to determine the extent of hydrogen dissolved in palladium. This knowledge is vital for applications in materials science and engineering where the hydrogen content can significantly affect the properties of the palladium metal. For instance, hydrogen storage and purification processes exploit the unique capabilities of palladium to absorb hydrogen.