Question

Solid thorium(IV) fluoride has a boiling point of \(1680^{\circ} \mathrm{C}\). What is the density of a sample of gaseous thorium(IV) fluoride at its boiling point under a pressure of \(2.5\) atm in a 1.7-L container? Which gas will effuse faster at \(1680^{\circ} \mathrm{C}\), thorium(IV) fluoride or uranium(III) fluoride? How much faster?

Short answer

The density of gaseous thorium(IV) fluoride at its boiling point under a pressure of $2.5$ atm in a $1.7$-L container is approximately \(7.64 \frac{g}{L}\). Thorium(IV) fluoride will effuse faster than uranium(III) fluoride at $1680^{\circ} \mathrm{C}$, about $1.21$ times faster.

Step-by-step solution

Convert temperature to Kelvin

To use the Ideal Gas Law, we must first convert the given temperature to Kelvin. To do this, we will add 273.15 to the given Celsius temperature. \[T_K = T_C + 273.15\]

Find moles of thorium(IV) fluoride

We will use the Ideal Gas Law, PV=nRT, to find moles (n) of thorium(IV) fluoride, where P is pressure, V is volume, R is the universal gas constant, and T is temperature. \[n = \frac{PV}{RT}\]

Calculate molar masses of thorium(IV) fluoride and uranium(III) fluoride

In order to find the mass of thorium(IV) fluoride and to compare the effusion rates, we need to calculate the molar masses of thorium(IV) fluoride and uranium(III) fluoride. Molar mass of Thorium(IV) Fluoride (ThF4): M(ThF4)=M(Th) + 4 * M(F) Molar mass of Uranium(III) Fluoride (UF3): M(UF3)=M(U) + 3 * M(F)

Find mass of thorium(IV) fluoride

Use the moles of thorium(IV) fluoride (from Step 2) and molar mass (from Step 3) to find the mass of the substance. \[mass = moles \times molar\ mass\]

Calculate the density of thorium(IV) fluoride

Using mass (from Step 4) and volume (given), we will calculate the density of thorium(IV) fluoride. \[density = \frac{mass}{volume}\]

Compare effusion rates of thorium(IV) fluoride and uranium(III) fluoride

We will use Graham's Law of Effusion to compare the effusion rates of thorium(IV) fluoride and uranium(III) fluoride at the boiling point of thorium(IV) fluoride (from Step 1). \[ \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} \] Where \(r_1\) and \(r_2\) are the effusion rates of two gases, and \(M_1\) and \(M_2\) are their molar masses (from Step 3).

Calculate how much faster one gas effuses than the other

Using the ratio of effusion rates (from Step 6), calculate how much faster one gas effuses than the other. \[ratio = \frac{r_1}{r_2} \]

Key concepts

Calculation of Gas Density

Understanding the density of a gas is instrumental in fields such as chemistry and engineering. Density is defined as mass per unit volume, encapsulated in the formula \( density = \frac{mass}{volume} \). When calculating the density of a gas, we can utilize the Ideal Gas Law \( PV = nRT \) where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

Firstly, we find the mass of the gas by multiplying the number of moles by its molar mass. Then, using the volume of the container holding the gas, we apply the aforementioned formula to find the gas's density. The process is a critical tool in predicting how a gas will behave under different conditions and aids in various practical applications such as estimating atmospheric pressure or designing chemical reactors.

Graham's Law of Effusion

Graham's Law of Effusion is a principle that describes the relationship between the rate of effusion for gases and their molar masses.

Effusion is a process where gas particles pass through a tiny opening from a container to a vacuum. According to Graham's Law, the rate of effusion (\( r \) of a gas is inversely proportional to the square root of its molar mass (\( M \) as expressed in the formula \( \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} \).

Therefore, if comparing two gases under the same conditions, the lighter gas (with a lower molar mass) will effuse more quickly than the heavier gas. This law provides crucial insight when separating isotopes or analyzing gaseous mixtures in fields ranging from industrial processes to environmental science.

Molar Mass Determination

Molar mass determination is a fundamental concept in chemistry that involves calculating the mass of one mole of a substance, typically expressed in grams per mole (g/mol). The molar mass is the sum of the atomic masses of all the atoms in a molecule. For instance, to find the molar mass of a compound like thorium(IV) fluoride (ThF4), we would sum the atomic mass of thorium (Th) with four times the atomic mass of fluorine (F).

Understanding this concept allows chemists to convert between the mass of a substance and the number of moles, a crucial step in many stoichiometric calculations. These calculations are vital for preparing solutions, scaling reactions accurately, and understanding the atomic composition of molecules.

Converting Celsius to Kelvin

Temperature conversion is an everyday necessity in science, especially when dealing with temperature-sensitive experiments such as gas reactions. The Kelvin (K) temperature scale is the base unit of thermodynamic temperature in the International System of Units (SI), and unlike Celsius (°C), it starts at absolute zero, the theoretically lowest temperature possible.

To convert Celsius to Kelvin, we add 273.15 to the Celsius temperature, which accounts for the difference in the starting points of the two scales. The equation \( T_K = T_C + 273.15 \) provides the conversion and is crucial in accurately applying laws like the Ideal Gas Law, which require temperature to be in Kelvin for proper calculation and analysis of gases' behaviors.