Question

Identify each variable in the following formula: $$\Delta T_{\mathrm{b}}=K_{\mathrm{b}} m$$

Short answer

In the formula \(\Delta T_{\mathrm{b}}=K_{\mathrm{b}} m\), \(\Delta T_{\mathrm{b}}\) represents the change in temperature due to boiling, \(K_{\mathrm{b}}\) is the boiling-point elevation constant, and \(m\) stands for the molality of the solute in the solution.

Step-by-step solution

Identify the first variable

The first variable in the formula is \(\Delta T_{\mathrm{b}}\). This represents the change in temperature in degrees Celsius or Kelvin due to boiling.

Identify the second variable

The second variable in the formula is \(K_{\mathrm{b}}\). This is known as the boiling-point elevation constant. It is an experimentally determined constant for a particular solvent, which describes how much the boiling point of a solution will increase in relation to the concentration of solute particles.

Identify the third variable

The third variable in the formula is \(m\). This variable represents the molality of the solute in the solution, typically expressed in moles solute per kilogram solvent.

Key concepts

Understanding Temperature Change

When we talk about temperature change in the context of chemistry, we specifically mean the difference between the initial and final temperature of a substance. In the formula \(\Delta T_{\mathrm{b}}=K_{\mathrm{b}} m\), \(\Delta T_{\mathrm{b}}\) symbolizes this temperature change due to the boiling point elevation phenomenon. It's crucial for students to grasp that this \(\Delta\) symbol (delta) refers to a change or difference in temperature. For instance, if water normally boils at 100°C, but with a solute dissolved in it boils at 102°C, the \(\Delta T_{\mathrm{b}}\) would be 2°C.

Temperature change is a vital concept in thermodynamics and physical chemistry because it can influence the behavior and properties of substances. In the context of boiling point elevation, it is the measurable outcome that results from adding a non-volatile solute to a solvent. The increased boiling point can affect various applications, from culinary techniques to industrial processes, making it essential to understand how and why this change occurs.

Decoding the Boiling-Point Elevation Constant

The boiling-point elevation constant, represented as \(K_{\mathrm{b}}\) in our equation, acts as a proportionality factor that relates the molality of the solute in a solution to the increase in the boiling point. This constant is specific to the solvent and must be determined experimentally. It represents the boiling point elevation of the solvent when a one molal solution (1 mol/kg) of a non-volatile solute is dissolved in it.

Why does each solvent have a unique \(K_{\mathrm{b}}\)?

The physical properties of each solvent, like intermolecular forces and the heat needed to vaporize it, play a role in why \(K_{\mathrm{b}}\) varies. For instance, water has a \(K_{\mathrm{b}}\) value of 0.512°C·kg/mol, which means for every 1 molal increase in the concentration of a solute, the boiling point of water is elevated by approximately 0.512°C. Understanding this constant is fundamental in calculating the expected temperature change and in turn, applying this knowledge to practical situations.

Molality and its Role in Boiling-Point Elevation

Molality, denoted as \(m\) in our equation, is a measure of the concentration of a solution. It is defined as the number of moles of solute per kilogram of solvent. Unlike molarity, which is volume-based, molality is mass-based and is not affected by temperature fluctuations, making it a more reliable unit of concentration in temperature-related calculations such as boiling point elevation.

How to calculate molality: It's a simple division of the number of moles of solute by the mass of the solvent in kilograms. As an example, if we dissolve 2 moles of salt (solute) in 1 kilogram of water (solvent), the molality of the solution is 2 mol/kg.

Practical implication of molality in boiling-point elevation:

Knowing the molality of a solution allows us to predict how much higher the boiling point will be. In real-world applications, this can range from determining the concentration of anti-freeze in a car's radiator to adjusting the cooking time of food at higher altitudes. It's an integral part of the calculation that works hand-in-hand with the boiling-point elevation constant to find the temperature change induced by a solute.