Dark streaks have been observed on the surface of Mars These streaks sometimes seem to run or flow, and other times they appear static. It has been hypothesized that these streaks are salt (sodium chloride) water rivers that are sometimes melted, allowing them to flow, and at other times these rivers are frozen. If the salt water mixtures melt when the temperature rises to \(-12.7^{\circ} \mathrm{C}\), what is the molality of salt in the rivers? (Hint: The freezing point is determined by the molality of ions. How will the molality of sodium chloride be related to the molality of the ions?)

Molality is a way to express the concentration of a solution. It is defined as the number of moles of solute per kilogram of solvent. Unlike molarity, which depends on volume, molality depends on the mass of the solvent and is not affected by temperature changes. This makes it a useful measure when dealing with temperature-dependent properties like freezing point depression.

To calculate molality, use the formula:

\( m = \frac{\text{moles of solute}}{\text{kilograms of solvent}} \).

In our exercise, we are given conditions where water is the solvent, and sodium chloride (\( \text{NaCl} \)) is the solute. By knowing the change in the freezing point, we can calculate the molality of the salt water mixture. The calculated molality tells us how concentrated the salt solution is.

The van't Hoff factor (\( i \)) is a dimensionless value that indicates the number of particles a solute splits into or produces when dissolved in a solvent. For ionic compounds like sodium chloride (\( \text{NaCl} \)), the van't Hoff factor is equal to the total number of ions the compound dissociates into.

For example, \( \text{NaCl} \) dissociates in water into \( \text{Na}^+ \) and \( \text{Cl}^- \). Thus, for \( \text{NaCl} \), \( i = 2 \) because it forms two ions.

In the given exercise, applying the van't Hoff factor helps us determine the effective concentration of particles in solution, which directly affects the freezing point depression. When we use the van't Hoff factor in the freezing point depression formula, we account for the increased number of particles.

The cryoscopic constant (\( K_f \)) is a property of the solvent that represents the freezing point depression per molal concentration of a solute. Each solvent has a unique \( K_f \) value which is determined experimentally.

For water, \( K_f \) is \( 1.86^{\text{C} \thinspace \text{kg/mol}} \). This means that for every 1 molal increase in concentration of a solute, the freezing point of water is lowered by \( 1.86 \thinspace \text{C} \).

In our calculations, we use this constant with the freezing point depression formula \( \Delta T_f = i K_f m \) to find out the molality of the salt in the solution. This constant allows us to translate the observed change in freezing point directly into the concentration of the solute.