A closed vessel system of volume 2.5 L contains a mixture of neon and fluorine. The total pressure is 3.32 atm at 0.0°C. When the mixture is heated to 15°C, the entropy of the mixture increases by 0.345 J/K. What amount (in moles) of each substance (Ne and F2) is present in the mixture? (heat capacity of monoatomic gas = 3/2R and that for a diatomic gas = 5/2R)

moles Ne = 0.154 mol

moles F₂ = 0.217 mol

Explanation:

Step 1: Data given

Volume of the vessel system = 2.5 L

Total pressure = 3.32 atm at 0.0 °C

The mixture is heated to 15.0 °C

The entropy of the mixture increases by 0.345 J/K

The heat capacity of monoatomic gas = 3/2R and that for a diatomic gas = 5/2R

Step 2: Define the gas

Neon is a monoatomic gas, composed of Ne atoms

⇒ Cv (Ne) ≅ (3/2) R

Fluorine is a diatomic gas, composed of F₂ molecules.

⇒ Cv (F₂) ≅ (5/2) R

Step 3: Calculate moles of gas

p*V = n*R*T

⇒ with p = the total pressure = 3.32 atm

⇒ with V = the total volume = 2.5 L

⇒ with n = the number of moles of gas

⇒ with R = the gas constant = 0.08206 L*atm/K*mol

⇒ with T = the temperature = 273.15 Kelvin

n (total) = p*V/RT = (3.32 atm*2.5 L) / (0.08206 L*atm/mol•K*273.15) = 0.3703 mol

Step 4: Calculate moles of Ne and F2

For one mole heated at constant volume,

∆S = Cv*ln (288.15/273.15) = 0.05346*Cv

⇒ ∆S for 0.3703 mol,

∆S = (0.3703 mol) (0.05346) Cv = 0.345 J/K

⇒ Cv = 17.43 J/mol*K for the Ne/F₂ mixture.

For pure Ne, Cv = (3/2) R = 1.5*8.314 J/mol*K = 12.471 J/mol*K

For pure F₂, Cv = (5/2) R = 2.5 * 8.314 J/mol*K = 20.785 J/mol*K

if X is the mole fraction of Ne, we can find X by:

17.43 J/mol*K = X * 12.471 J/mol*K + (1 - X) * 20.785 J/mol*K

⇒ 20.875 - 8.314 * X = 17.43

X = 0.415, 1 - X = 0.585

moles Ne = (0.415) (0.3703 mol) = 0.154 mol

moles F₂ = (0.585) (0.3703 mol) = 0.217 mol