Problem 74 - Chemistry The Molecular Science

Question

A 425-gal tank of water contains \(175 \mathrm{~g}\) NaI. Calculate the volume (L) of chlorine gas at \(758 \mathrm{mmHg}\) and \(25^{\circ} \mathrm{C}\) required to convert all the iodide to iodine.

Step-by-Step Solution

Answer

14.38 liters of chlorine gas are required.

Step 1: Determine the moles of NaI

First, find the moles of Sodium Iodide (NaI) in the 425-gallon tank. The molar mass of NaI is the sum of the atomic masses of Na (22.99 g/mol) and I (126.90 g/mol), which is 149.89 g/mol. Thus, the number of moles of NaI is calculated as:\[moles \, NaI = \frac{175 \, g}{149.89 \, g/mol} \approx 1.167 \, mol\]

Step 2: Moles of Chlorine Needed

Since each mole of Cl extsubscript{2} reacts with 2 moles of NaI, the stoichiometry of the reaction is:\[2 \text{NaI} + \text{Cl}_2 \rightarrow 2 \text{NaCl} + \text{I}_2\]This means you need half the moles of Cl extsubscript{2} compared to NaI to react completely. Therefore, the moles of Cl extsubscript{2} required are:\[moles \, Cl_2 = \frac{1.167 \, mol}{2} \approx 0.5835 \, mol\]

Step 3: Convert Pressure to Atmospheres

Before using the ideal gas law, convert the pressure from mmHg to atm, noting that 1 atm = 760 mmHg:\[P = \frac{758 \, mmHg}{760 \, mmHg/atm} \approx 0.9974 \, atm\]

Step 4: Use the Ideal Gas Law

Apply the ideal gas law \(PV = nRT\) to find the volume of Cl extsubscript{2} gas needed. Use \(R = 0.0821 \, L\cdot atm/K\cdot mol\), and convert the temperature to Kelvin by adding 273.15 to the Celsius temperature:\[T = 25^{\circ}C + 273.15 = 298.15 \, K\]Plug the values into the ideal gas law:\[V = \frac{nRT}{P} = \frac{0.5835 \, mol \times 0.0821 \, L\cdot atm/K\cdot mol \times 298.15 \, K}{0.9974 \, atm} \approx 14.38 \, L\]

Step 5: Final Answer

The volume of chlorine gas required to convert all the iodide to iodine under the given conditions is approximately 14.38 liters.

Key Concepts

Stoichiometry

Stoichiometry is like the recipe of chemistry that helps you understand the proportions in which reactants combine and products form in a chemical reaction. For this exercise, we're looking at the reaction between sodium iodide (NaI) and chlorine gas (\(\text{Cl}_2\)) to form sodium chloride (NaCl) and iodine (\(\text{I}_2\)).

Here's how it works:

• Each mole of \(\text{Cl}_2\) needs two moles of NaI to react with.
• The balanced chemical equation is \[2 \text{NaI} + \text{Cl}_2 \rightarrow 2 \text{NaCl} + \text{I}_2\]
• Knowing this stoichiometry, if you have 1.167 moles of NaI, you need half as many moles of \(\text{Cl}_2\), i.e., 0.5835 moles.

So, stoichiometry helps us calculate the exact quantities of reactants required, ensuring no waste or excess.

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry that describes the behavior of an ideal gas. The equation is \(PV = nRT\), where:

• \(P\) stands for pressure.
• \(V\) is volume.
• \(n\) is the number of moles of the gas.
• \(R\) is the gas constant—0.0821 \(L\cdot atm/K\cdot mol\).
• \(T\) is the temperature in Kelvin.

To find the volume of chlorine gas needed in this exercise, the ideal gas law provides the means:

• Convert the Celsius temperature (25°C) to Kelvin: \(T = 298.15\, K\).
• Plug in the values: moles (\(0.5835\)), R, T, and the pressure in atm, to find volume.

Using the formula -\[V = \frac{nRT}{P} \approx 14.38 \text{ liters}\] - we can calculate the volume needed accurately.

Pressure Conversion

Pressure conversion is necessary when dealing with gas laws as values of pressure can be in different units. In our example, the pressure is given in mmHg, but we converted it to atmospheres (atm) since the ideal gas constant \(R\) we used is in units compatible with atm.

Here's the conversion process:

• 1 atm is equal to 760 mmHg.
• To convert any given pressure in mmHg to atm, divide the mmHg value by 760.

For this problem, converting 758 mmHg to atm:\[P = \frac{758 \, \text{mmHg}}{760 \, \text{mmHg/atm}} \approx 0.9974 \, \text{atm}\]This conversion simplifies problems involving gases by using consistent units, making subsequent calculations straightforward and accurate.