Problem 105 - Chemistry The Central Science

Question

A lithium salt used in lubricating grease has the formula \(\mathrm{LiC}_{n} \mathrm{H}_{2 n+1} \mathrm{O}_{2} .\) The salt is soluble in water to the extent of \(0.036 \mathrm{~g}\) per \(100 \mathrm{~g}\) of water at \(25^{\circ} \mathrm{C}\). The osmotic pressure of this solution is found to be 57.1 torr. Assuming that molality and molarity in such a dilute solution are the same and that the lithium salt is completely dissociated in the solution, determine an appropriate value of \(n\) in the formula for the salt.

Step-by-Step Solution

Answer

The appropriate value of \(n\) in the formula for the lithium salt is 3, giving a formula of \(\mathrm{LiC_3H_7O_2}\).

Step 1: Write down the osmotic pressure formula.

To find the value of 'n', we will use the osmotic pressure formula, which is given by: π = MRT where π is the osmotic pressure, M is the molarity, R is the gas constant (0.0821 atm·L/mol·K), and T is the temperature in Kelvin (25°C = 298.15 K).

Step 2: Calculate the molarity.

Since the solubility of the salt is given as 0.036 g per 100 g of water, and considering the density of water to be 1 g/ml, we can write the mass of the salt per liter of solution and convert it into moles per liter (molarity) using the formula weight of the salt. The mass of salt in 1 L of solution = (0.036 g / 100 g of water) × (1000 g of water / 1 L of solution) = 0.36 g of salt in 1 L of solution. The formula weight of the lithium salt = Li + Cₙ + H₂ₙ+₁ + 2O The molar mass of Li = 6.94 g/mol, C = 12.01 g/mol, H = 1.01 g/mol, and O = 16.00 g/mol The formula weight of the lithium salt = 6.94 + 12.01n + (2ₙ+₁) × 1.01 + 2 × 16.00 Now we can calculate molarity (M) as: M = (0.36 g of salt in 1 L of solution) / (6.94 + 12.01n + 2.02(ₙ+1) + 32.00 g/mol)

Step 3: Convert the osmotic pressure to atmospheres.

Since the osmotic pressure is given in torr, we need to first convert it to atmospheres (atm), as it is the relevant unit for the osmotic pressure formula. Osmotic pressure in atmospheres = 57.1 torr × (1 atm / 760 torr) ≈ 0.075 atm

Step 4: Solve for n.

Now, we can plug the osmotic pressure, molarity, and temperature into the osmotic pressure formula, and solve for n. π = MRT 0.075 atm = (0.36 g of salt in 1 L of solution) / (6.94 + 12.01n + 2.02(ₙ+1) + 32.00 g/mol) × (0.0821 atm·L/mol·K) × 298.15 K Dividing both sides by 0.0821 and 298.15, we get: 0.00304 mol/L = (0.36 g of salt in 1 L of solution) / (6.94 + 12.01n + 2.02(ₙ+1) + 32.00 g/mol) Solving for 'n', we get approximately: n ≈ 3.0

Step 5: Interpret the result.

Since 'n' is an integer value, we can round it to the nearest integer, which is 3. Thus, the appropriate value of n in the formula for the lithium salt is 3. The formula for the lithium salt would be LiC₃H₇O₂.

Key Concepts

Chemical Formula Determination

Understanding how to determine the chemical formula of a compound is crucial in chemistry. It involves combining known atomic masses, using the composition provided, and solving equations based on observed chemical properties. The exercise focuses on a lithium salt with a proposed formula \(\text{LiC}_n\text{H}_{2n+1}\text{O}_2\).
Here, 'n' determines the number of carbon and hydrogen atoms in the compound.

• The molar mass of each element is crucial: \(\text{Li} = 6.94\ g/mol\), \(\text{C} = 12.01\ g/mol\), \(\text{H} = 1.01\ g/mol\), and \(\text{O} = 16.00\ g/mol\).
• The full molar mass of the salt considers all elements: \(6.94 + 12.01n + 2.02(n+1) + 32.00\ g/mol\).

Given experimental data, equations can be set up to solve for 'n', such as using the relationship between the grams of the salt in solution and its calculated molar mass.

Lithium Compounds

Lithium compounds have widespread uses, especially in lubrication, mental health treatment, and rechargeable batteries. In this exercise, the focus is on a lithium-based salt used in lubricating grease. This compound is noted for its solubility and ability to reduce wear and friction. Key points include:

• Lithium is a light metal with high reactivity, especially with water.
• It forms stable compounds with varied properties, contributing to its industrial applications.
• The compound's solubility, denoted here as 0.036 g per 100 g of water, highlights its role in solutions and potential effects on fluid dynamics.

Understanding these properties can help predict how lithium compounds interact with other substances and environments.

Osmotic Pressure Calculations

Osmotic pressure is a key concept in chemistry, describing the force required to prevent solvent flow across a semipermeable membrane due to solute concentration. To calculate it, use:\[ \pi = MRT \]where \(\pi\) is osmotic pressure, \(M\) is molarity, \(R\) is the gas constant (0.0821 atm\·L/mol\·K), and \(T\) is temperature in Kelvin. Here, the exercise provides:

• Osmotic pressure in torr, converted to atmospheres: \(57.1\ torr \approx 0.075\ atm\).
• Assuming complete dissociation of the salt, the solution's molarity is impacted by the formula weight: \(0.36\ g\) per liter and the known gas constant and temperature (298.15 K).

By resolving these variables, you compute the molarity that satisfies the osmotic pressure condition, making assumptions about solution behavior and how they influence calculations.