Question

An experiment calls for \(1.00 \mathrm{g}\) of silver nitrate, but all that is available in the laboratory is a \(0.50 \%\) solution of \(\mathrm{AgNO}_{3}\). Assuming the density of the silver nitrate solution to be very nearly that of water because it is so dilute, determine how many milliliters of the solution should be used.

Short answer

To obtain \(1.00 \mathrm{g}\) of AgNO\(_3\) from a \(0.50 \%\) solution, first calculate the mass of AgNO\(_3\) per 100g of solution: \(100 \times 0.50 \% = 0.50\mathrm{g}\). Then, set up a proportion to find the mass of the solution needed to get 1g of AgNO\(_3\): \(\frac{1}{x}\:=\:\frac{0.50}{100}\). Solve for x: \(x = \frac{1 \times 100}{0.50} = 200\mathrm{g}\). Since the density of the solution is assumed to be that of water (1g/mL), the volume of the solution needed is equal to the mass of the solution needed (200g). Therefore, \(200\mathrm{mL}\) of the \(0.50 \%\) AgNO\(_3\) solution should be used to obtain \(1.00 \mathrm{g}\) of silver nitrate.

Step-by-step solution

Calculating the mass of AgNO\(_3\) in the solution

Since the solution is \(0.50 \%\) AgNO\(_3\), we find the mass of AgNO\(_3\) in a 100g silver nitrate solution. Multiply the total mass of the solution (100g) by the mass percentage of AgNO\(_3\): mass of AgNO\(_3\) per 100g of solution = \(100 \times 0.50 \%\)

Calculating the mass of the solution required to get 1g of AgNO\(_3\)

Since we need 1g of AgNO\(_3\) for the experiment, we can calculate the mass of the solution required to get that amount of AgNO\(_3\). We can use a simple proportion to solve for this: \(\frac{1}{x}\:=\:\frac{0.50}{100}\) where x represents the mass of the solution needed to get 1g of AgNO\(_3\). Solve for x: x = \(\frac{1 \times 100}{0.50}\)

Calculate the volume of the solution needed

Since we assume that the density of the AgNO\(_3\) solution is nearly that of water, we can use the density of water (1g/mL) to calculate the volume of the solution needed. 1g of water is equivalent to 1 mL of water. Volume of the AgNO\(_3\) solution needed (in mL) = mass of the AgNO\(_3\) solution needed (in g) Now, find the volume: Volume in mL = x (Calculated in Step 2)

Key concepts

Mass Percentage

When you're working with solutions, mass percentage is a key concept. It tells you how much of a solute is present in a given mass of solution. Mass percentage is calculated as:\[ \text{Mass Percentage} = \left( \frac{\text{Mass of Solute}}{\text{Total Mass of Solution}} \right) \times 100 \%\]In the given exercise, a silver nitrate (\(\mathrm{AgNO}_3\)) solution has a concentration of 0.50%. This indicates that in every 100 grams of the solution, 0.50 grams is silver nitrate. This is a convenient way to express concentration, especially in laboratory settings. It provides a clear understanding of the solute's presence relative to the entire solution mass. Understanding mass percentage helps in measuring specific amounts of substance required for experiments.

Density Assumptions

When working with dilute solutions, it's common to assume the solution has the same density as water, which is 1 g/mL. This assumption simplifies calculations without significantly affecting the accuracy for very dilute solutions. Since water and the dilute silver nitrate solution have close densities, this assumption helps us convert between mass and volume directly. In the exercise, the density assumption allows us to equate the mass (in grams) of the silver nitrate solution to its volume (in mL), making it easier to measure in laboratory equipment. Density assumptions are particularly useful when preparing solutions where precision does not need to be extremely high.

Silver Nitrate Solution

Silver nitrate is a common chemical used in laboratories. It is denoted by the formula \( \mathrm{AgNO}_3 \) and typically found as a white crystalline solid. When dissolved in water, it forms a solution used in a variety of experiments. For practical applications, AgNO\(_3\) solutions are often prepared in specific concentrations, like the 0.50% solution in the exercise. This low concentration means the solution is very dilute, which is why density assumptions can be validly applied. Understanding how to calculate the needed volume of a silver nitrate solution is essential for preparing precise experimental conditions.

Proportion Method

The proportion method is a handy mathematical approach used to determine the necessary amounts in solution-based calculations. In chemical calculations like the given exercise, you use proportions to relate different quantities. The proportion method involves setting up an equation that relates the known concentration with the desired outcome:\[ \frac{ \text{Amount needed}}{ \text{Amount available}} = \frac{ \text{Mass percentage}}{ 100 }\]In this exercise, to find out how many grams of the silver nitrate solution are needed to get 1 gram of AgNO\(_3\), a proportion is set up:\( \frac{1}{x} = \frac{0.50}{100} \)Solving this gives you the mass of the solution needed. This method simplifies the process of working out exact amounts when dealing with solutions, making it straightforward to adjust and find precise quantities needed in different scenarios.