Question

Aqueous ammonia is typically sold by chemical supply houses as the saturated solution, which has a concentration of \(14.5 \mathrm{~mol} / \mathrm{L}\). What volume of \(\mathrm{NH}_{3}\) at STP is required to prepare 100 . \(\mathrm{mL}\) of concentrated ammonia solution?

Short answer

To prepare 100 mL of concentrated ammonia solution with a concentration of 14.5 mol/L, we need to find the volume of NH3 gas at STP. First, we convert the volume to liters (0.100 L), then calculate the moles of NH3 needed (1.45 mol). We apply the Ideal Gas Law (PV = nRT) to solve for the volume of NH3 gas at STP, which is 32.47 L. Therefore, we require 32.47 L of NH3 gas at STP to prepare 100 mL of concentrated ammonia solution.

Step-by-step solution

Convert volume of concentrated ammonia solution to liters

First, convert the given volume (100 mL) of concentrated ammonia solution to liters. 100 mL * (1 L / 1000 mL) = 0.100 L

Calculate moles of NH3 needed

Using the given concentration of the saturated solution (14.5 mol/L), calculate the moles of NH3 needed for the 100 mL of concentrated ammonia solution. Moles of NH3 = Concentration * Volume Moles of NH3 = 14.5 mol/L * 0.100 L Moles of NH3 = 1.45 mol

Use the Ideal Gas Law to find the volume of NH3 gas at STP

We have the number of moles of NH3, so we can apply the Ideal Gas Law (PV = nRT) to find the volume of the NH3 gas at STP (Standard Temperature and Pressure: P = 1 atm, T = 273 K, R = 0.0821 L*atm/mol*K), and solve for V: PV = nRT V = nRT / P V = (1.45 mol) * (0.0821 L*atm/mol*K) * (273 K) / (1 atm) V = 32.47 L

Convert the volume to the appropriate unit

Now, we have the volume of NH3 gas at STP required, which is 32.47 L. Our final answer is 32.47 L. The student should report this as their final answer if asked for the volume of NH3 gas, to prepare 100 mL of concentrated ammonia solution at STP.

Key concepts

Molarity Calculation

Molarity is a term used in chemistry to specify the concentration of a solution. It is defined as the number of moles of solute (the substance being dissolved) per liter of solution. The formula for calculating molarity (M) is:
\( M = \frac{\text{moles of solute}}{\text{liters of solution}} \).
This concept is pivotal when preparing solutions in the lab, as it allows scientists to accurately convey the strength of a solution. In the context of preparing an aqueous ammonia solution, knowing the molarity allows us to determine the exact amount of ammonia (in moles) required for a certain volume of solution. For instance, given a solution concentration of 14.5 mol/L, we need to multiply this molarity by the volume of the solution we want to prepare in liters to find out the moles of ammonia needed.
In practice, the molarity calculation is straightforward, just as seen in the step by step solution where 100 mL (or 0.100 L) of concentrated ammonia solution at a molarity of 14.5 mol/L requires 1.45 moles of ammonia. It’s essential to grasp that molarity is directly proportional to the amount of solute in a given volume of solution; as the molarity increases, the more solute (or in this case, ammonia) is present per unit of volume.

Gas Volume Calculation

The gas volume calculation is a technique used to determine the volume occupied by a gas under specific conditions, usually expressed in liters. For gases, volume can significantly change with adjustments in temperature and pressure, which is why Standard Temperature and Pressure (STP) is often used as a reference point for calculations. STP is defined as 0 degrees Celsius (273 K) and 1 atmosphere (atm) of pressure.
When dealing with gases like ammonia, calculating the volume of gas requires understanding the relationship between moles, pressure, temperature, and volume. At STP, one mole of any ideal gas occupies 22.4 liters. Therefore, if we know the number of moles, we can easily calculate the volume at STP by multiplying the moles by 22.4 liters/mole. Moreover, if the conditions are not at STP, one can still find the volume using the Ideal Gas Law, establishing a proportional relationship between these variables.
In the given exercise, we aim to find out how much volume of ammonia gas at STP corresponds to the moles needed for the solution. As seen in the step by step solution, using the Ideal Gas Law, the calculated volume for the required moles of ammonia at STP turns out to be 32.47 liters.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics that relates the pressure (P), volume (V), temperature (T), and number of moles (n) of an ideal gas through the equation \( PV = nRT \),
where R is the universal gas constant, which has a value of 0.0821 L*atm/mol*K when pressure is measured in atmospheres and volume in liters. This law is based on the assumption that the gas particles are in constant motion and do not interact with each other, which is a good approximation for many real-world scenarios.
The equation can be manipulated to solve for any one of the variables if the others are known. This is particularly useful in chemistry when working with reactions involving gases, or preparing solutions from gases, as it links the macroscopic properties we can measure (P, V, and T) with the microscopic property we often need to know (n). In the exercise provided, we use the Ideal Gas Law to calculate the volume of ammonia gas at STP needed for the solution. By inserting the known values for n, R, and T (at STP conditions), and rearranging the formula to solve for V, we find out the volume of gas required.