Problem 30 - Chemistry: Principles and Reactions

Question

A \(2.00-g\) sample of \(S X_{6}(g)\) has a volume of \(329.5 \mathrm{~cm}^{3}\) at \(1.00 \mathrm{~atm}\) and \(20^{\circ} \mathrm{C}\). Identify the element \(\mathrm{X}\). Name the compound.

Step-by-Step Solution

Answer

Answer: The element X is chlorine (Cl), and the compound is sulfur hexachloride (SCL₆).

Step 1: Convert the given values to appropriate units

First, we need to convert the given values to appropriate units. Volume: Convert the volume from cm³ to liters: 1 L = 1000 cm³ 329.5 cm³ = 329.5 / 1000 L = 0.3295 L Temperature: Convert the temperature from Celsius to Kelvin: T(K) = T(°C) + 273.15 T(K) = 20 °C + 273.15 = 293.15 K Now, we have: P = 1.00 atm V = 0.3295 L T = 293.15 K

Step 2: Use the Ideal Gas Law to find the number of moles of the given sample

We will use the Ideal Gas Law equation PV = nRT to find the number of moles, n, of the given gaseous sample. We can rearrange the equation to solve for moles as follows: n = PV/(RT) Now, plug in the values: n = (1.00 atm)(0.3295 L) / (0.0821 L atm / K mol)(293.15 K) n = 0.0148 moles

Step 3: Calculate the molar mass of the compound

We know the mass of the sample and the number of moles, so we can calculate the molar mass of the compound: Molar mass (M) = mass (g) / moles (n) M = 2.00 g / 0.0148 moles M = 135.14 g/mol

Step 4: Divide the molar mass of the compound by the molar mass of sulfur to get the molar mass of the element X

The chemical formula of the compound is \(S X_{6}\). So, to find the molar mass of element X, divide the total molar mass of the compound by the molar mass of sulfur. Molar mass of sulfur, \(S= 32.07 g/mol\) Molar mass of \(6X = 135.14 g/mol - 32.07 g/mol = 103.07 g/mol\) Molar mass of element X = 103.07 g/mol ÷ 6 Molar mass of element X = 17.18 g/mol

Step 5: Identify the element X and name the compound

The element with a molar mass of approximately 17.18 g/mol is chlorine (Cl). Thus, the compound is sulfur hexachloride (SCL₆). Compound name: Sulfur hexachloride.

Key Concepts

Molar Mass Calculation

Molar mass calculation is a pivotal skill in chemistry. It allows us to determine how much one mole of a particular substance weighs. To calculate the molar mass, you need the mass of the sample and the number of moles it contains. In the provided exercise, the gas sample has a mass of 2.00 grams and is comprised of 0.0148 moles, which we find using the Ideal Gas Law. Dividing these two values gives the molar mass of the compound:

\[ \text{Molar Mass} (M) = \frac{\text{mass (g)}}{\text{moles (n)}} \]
Using the numbers from the exercise, the calculation yields 135.14 g/mol. Knowing how to perform these calculations helps in identifying unknown compounds and elements in chemical formulas.

Gas Properties

Gases exhibit unique properties governed by the Ideal Gas Law, which is expressed as \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is temperature in Kelvin.
Understanding this is crucial because it connects various physical properties of gases. The law illustrates that, under constant conditions, the pressure of a gas is directly proportional to its temperature and volume. It’s also inverse to the number of moles.

• This principle allows chemists to predict how gases will behave under different conditions.
• It helps determine quantities for reactions involving gaseous reactants or products.

For instance, in the exercise, converting temperature to Kelvin and volume to liters ensures accuracy in applying the Ideal Gas Law, ultimately aiding in calculating the number of moles present.

Chemical Formula Identification

Identifying chemical formulas involves understanding the molecule's composition, which sometimes requires determining unknown elements. In this process, we calculated the molar mass of the compound to discern its components.

In our exercise, after finding the molar mass of compound \(S X_6\) being 135.14 g/mol, we subtracted the known molar mass of sulfur (32.07 g/mol) from it.This difference corresponds to the molar mass of six of the unknown element "X". Dividing this molar mass difference by 6 gives the molar mass of X.

The calculated value for \(X\)'s molar mass was approximately 17.18 g/mol, which matches chlorine's molar mass. Thus, we concluded our compound was sulfur hexachloride \((SCl_6)\).
These steps emphasize the importance of molar mass and chemical formula knowledge in solving chemical mystery puzzles.