Consider the sublimation of mothballs at 27 °C in a room having dimensions 8.0 ft ⨉ 10.0 ft ⨉ 8.0 ft. Assume that the mothballs are pure solid naphthalene (density 1.16 g/cm³) and that they are spheres with a diameter of 12.0 mm. The equilibrium constant K_c for the sublimation of naphthalene is 5.40⨉10^-6 at 27 °C. C₁₀H₈(s) ⇌ C₁₀H₈(g) (a) When excess mothballs are present, how many gaseous naphthalene molecules are in the room at equilibrium?

Verified step by step guidance

1. Calculate the volume of the room in cubic meters by converting the dimensions from feet to meters and then multiplying them together.
2. Determine the volume of a single mothball using the formula for the volume of a sphere, \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius in meters.
3. Calculate the mass of a single mothball using its volume and the given density of naphthalene.
4. Use the equilibrium constant \( K_c \) to find the concentration of gaseous naphthalene at equilibrium. Since \( K_c = [C_{10}H_8(g)] \), the concentration is directly given by \( K_c \).
5. Convert the concentration of gaseous naphthalene to the number of molecules using the ideal gas law \( PV = nRT \) and Avogadro's number, considering the volume of the room and the conditions given.

Key Concepts

Sublimation

Sublimation is the phase transition in which a substance changes directly from a solid to a gas without passing through the liquid phase. In the context of naphthalene, this process occurs when solid mothballs release gaseous naphthalene molecules into the air. Understanding sublimation is crucial for analyzing how the concentration of naphthalene vapor reaches equilibrium in the given room.

Equilibrium Constant (Kc)

The equilibrium constant (Kc) is a numerical value that expresses the ratio of the concentrations of products to reactants at equilibrium for a reversible reaction. For the sublimation of naphthalene, Kc indicates the extent to which naphthalene vapor is present in the gas phase compared to the solid phase. A low Kc value, like 5.40×10^-6, suggests that at equilibrium, the concentration of gaseous naphthalene is much lower than that of the solid.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. This law is essential for calculating the number of gaseous naphthalene molecules in the room at equilibrium. By knowing the volume of the room and the temperature, one can use the equilibrium concentration derived from Kc to find the total number of gas molecules present.