Question

The solubility of lead nitrate at \(100^{\circ} \mathrm{C}\) is \(140.0 \mathrm{~g} / 100 \mathrm{~g}\) water. A solution at \(100^{\circ} \mathrm{C}\) consists of \(57.0 \mathrm{~g}\) of lead nitrate in \(64.0 \mathrm{~g}\) of water. When the solution is cooled to \(10^{\circ} \mathrm{C}, 25.0 \mathrm{~g}\) of lead nitrate crystallize out. What is the solubility of lead nitrate in \(g / 100 \mathrm{~g}\) water at \(10^{\circ} \mathrm{C} ?\)

Short answer

Answer: The solubility of lead nitrate at \(10^{\circ} \mathrm{C}\) is \(50 \mathrm{~g/100~g}\) water.

Step-by-step solution

Write down the given data

The solubility at \(100^{\circ} \mathrm{C}\) is \(140 \mathrm{~g/100~g}\) water, the mass of lead nitrate is \(57.0 \mathrm{~g}\), and the mass of water is \(64.0 \mathrm{~g}\). When cooled to \(10^{\circ} \mathrm{C}\), \(25.0 \mathrm{~g}\) of lead nitrate crystallize out.

Determine the initial mass of the solution

The initial mass of the solution at \(100^{\circ} \mathrm{C}\) is given by: $$ m_{solution} = m_{lead \thinspace nitrate} + m_{water} = 57.0 \mathrm{~g} + 64.0 \mathrm{~g} = 121.0 \mathrm{~g} $$

Calculate the mass of lead nitrate and water at \(10^{\circ} \mathrm{C}\)

When the solution is cooled to \(10^{\circ} \mathrm{C}\), \(25.0 \mathrm{~g}\) of lead nitrate crystallize out. So, the remaining mass of lead nitrate in the solution will be: $$ m_{lead \thinspace nitrate}^{10^{\circ}C} = m_{lead \thinspace nitrate} - m_{crystallized} = 57.0 \mathrm{~g} - 25.0 \mathrm{~g} = 32.0 \mathrm{~g} $$ The mass of water remains unchanged as it was at \(100^{\circ} \mathrm{C}\), which is \(64.0 \mathrm{~g}\).

Calculate the solubility of lead nitrate at \(10^{\circ} \mathrm{C}\)

Solubility is given as grams of solute per \(100 \mathrm{~g}\) of water. Thus, we can write the solubility at \(10^{\circ} \mathrm{C}\) as: $$ solubility = \frac{m_{lead \thinspace nitrate}^{10^{\circ}C}}{m_{water}} \times 100 $$ Plug in the values for the mass of lead nitrate at \(10^{\circ} \mathrm{C}\) and the mass of water: $$ solubility = \frac{32.0 \mathrm{~g}}{64.0 \mathrm{~g}} \times 100 = 50 \mathrm{~g/100~g} \thinspace water $$ So, the solubility of lead nitrate at \(10^{\circ} \mathrm{C}\) is \(50 \mathrm{~g/100~g}\) water.

Key concepts

Understanding Lead Nitrate

Lead nitrate is a white crystalline compound with the chemical formula \(\text{Pb(NO}_3\text{)}_2\). It is used in various industries, including manufacturing, pyrotechnics, and laboratories.

One key property of lead nitrate is its ability to dissolve in water, forming a solution. This characteristic is useful in many applications but also requires careful handling due to lead's toxic nature.

When discussing the solubility of lead nitrate, it is important to consider the temperature and the medium (in this case, water) it dissolves in. As the temperature changes, so does the solubility, affecting how much lead nitrate can be dissolved before reaching a saturation point where no more can dissolve.

In a typical lab setting or even a classroom experiment, understanding these principles is vital for predicting and measuring how lead nitrate interacts with water across different temperatures.

Temperature Effects on Solubility

Temperature plays a crucial role in the solubility of substances, including lead nitrate. As temperature increases, the solubility of many solids in water generally increases. However, this isn't universally true for all substances.

For lead nitrate, the solubility at \(100^{\circ} \mathrm{C}\) is much higher than at \(10^{\circ} \mathrm{C}\). This means more lead nitrate can dissolve in water at higher temperatures.

• High Temperature: Increased solubility allows more lead nitrate to dissolve.
• Low Temperature: Decreased solubility leads to more crystallization.

This temperature dependence is utilized in crystallization processes, where a solution is cooled to precipitate dissolved substances. Understanding how temperature affects solubility helps in predicting the amount of a solute that can remain in solution at different temperatures and is essential for various chemical and industrial processes.

The Process of Crystallization

Crystallization is a technique used to solidify substances from a solution. When a solution of lead nitrate is cooled, its solubility decreases, causing some of the dissolved compound to revert to its crystalline form.

At a higher temperature, more of the solid dissolves in water. Upon cooling, the reduction in solubility results in excess solute being deposited in the form of crystals. This effect is harnessed to purify substances or recover solids from solutions.

The process can be observed when \(25.0 \mathrm{~g}\) of lead nitrate crystallizes out when the solution is cooled from \(100^{\circ} \mathrm{C}\) to \(10^{\circ} \mathrm{C}\). This occurs because the solubility at \(10^{\circ} \mathrm{C}\) is lower, allowing less of the solute to stay dissolved, thus forming crystals.

Being able to control the temperature and understand the solubility at different levels enables chemists to efficiently carry out crystallization methods in both experimental and industrial settings.