Which of the following has the largest particle density (molecules per unit volume)?

(A) 0.8 L of nitrogen gas at 350 K and 100 kPa

(B) 1.0 L hydrogen gas at 350 K and 150 kPa

(C) 1.5 L oxygen gas at 300 K and 80 kPa

(D) 2.0 L of helium gas at 300 K and 120 kPa

To determine which option has the largest particle density, we need to calculate the number of molecules per unit volume for each gas sample using the ideal gas law, which is expressed as PV = nRT. Here, P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature in Kelvin. The particle density can be derived from the number of moles and the volume.

Step-by-Step Calculation

First, we will calculate the number of moles (n) for each gas using the ideal gas law rearranged to n = PV / RT. Then, we will find the particle density by dividing the number of moles by the volume (in liters).

Constants and Values

• R (ideal gas constant) = 8.314 J/(mol·K) = 0.0821 L·atm/(mol·K)
• 1 kPa = 0.00987 atm

Calculations for Each Option

Option A: 0.8 L of Nitrogen Gas at 350 K and 100 kPa

Convert pressure to atm: 100 kPa = 100 x 0.00987 = 0.987 atm.

Using the ideal gas law:

n = (P * V) / (R * T) = (0.987 atm * 0.8 L) / (0.0821 L·atm/(mol·K) * 350 K) = 0.027 moles.

Particle density = n / V = 0.027 moles / 0.8 L = 0.03375 moles/L.

Option B: 1.0 L of Hydrogen Gas at 350 K and 150 kPa

Convert pressure to atm: 150 kPa = 150 x 0.00987 = 1.4805 atm.

n = (1.4805 atm * 1.0 L) / (0.0821 L·atm/(mol·K) * 350 K) = 0.051 moles.

Particle density = 0.051 moles / 1.0 L = 0.051 moles/L.

Option C: 1.5 L of Oxygen Gas at 300 K and 80 kPa

Convert pressure to atm: 80 kPa = 80 x 0.00987 = 0.7896 atm.

n = (0.7896 atm * 1.5 L) / (0.0821 L·atm/(mol·K) * 300 K) = 0.048 moles.

Particle density = 0.048 moles / 1.5 L = 0.032 moles/L.

Option D: 2.0 L of Helium Gas at 300 K and 120 kPa

Convert pressure to atm: 120 kPa = 120 x 0.00987 = 1.1834 atm.

n = (1.1834 atm * 2.0 L) / (0.0821 L·atm/(mol·K) * 300 K) = 0.096 moles.

Particle density = 0.096 moles / 2.0 L = 0.048 moles/L.

Comparing Particle Densities

Now, let's summarize the particle densities calculated:

• Option A: 0.03375 moles/L
• Option B: 0.051 moles/L
• Option C: 0.032 moles/L
• Option D: 0.048 moles/L

From the calculations, the option with the largest particle density is Option B: 1.0 L of hydrogen gas at 350 K and 150 kPa, with a density of 0.051 moles/L.