An ideal gas composed of a mixture of oxygen and nitrogen is at a pressure of 1.50 atmospheres and a temperature of 25.0°C. What is the molar density? A) 79.3 mol/m³ B) 61.3 mol/m³ C) 55.5 mol/m³ D) 47.6 mol/m³ E) 40.0 mol/m³

Question

Quizplus · Accepted Answer

The molar density of an ideal gas is given by the formula:
n/V = P/(RT), where n is the number of moles, V is the volume, P is the pressure, R is the universal gas constant, and T is the temperature.

Since the gas is a mixture of oxygen and nitrogen, we can use the mole fraction of each component to calculate the molar density. Let xO2 be the mole fraction of oxygen and xN2 be the mole fraction of nitrogen. We know that xO2 + xN2 = 1 (since the total mole fraction must add up to 1).

We can use the partial pressures of O2 and N2 to calculate their mole fractions. According to Dalton's law of partial pressures, the total pressure of the mixture is the sum of the partial pressures of each component. Let PO2 and PN2 be the partial pressures of O2 and N2, respectively. We have:

PO2 + PN2 = Ptotal = 1.50 atm

The mole fraction of O2 is then given by:

xO2 = PO2/Ptotal

Similarly, the mole fraction of N2 is given by:

xN2 = PN2/Ptotal

We need to know the partial pressures of O2 and N2. To do this, we can assume that both gases behave as ideal gases and use the ideal gas law:

PV = nRT

For a given mass of gas, we can write:

n = m/M

where n is the number of moles, m is the mass, and M is the molar mass. We can rearrange the ideal gas law to solve for the partial pressure:

P = (m/M)(RT/V)

For O2, we have:

MO2 = 32.00 g/mol

mO2/V = dO2 = nO2/V(MO2) = xO2n/V(MO2)

PO2 = xO2Ptotal = (mO2/V)(RT/MO2) = dO2RT(Ptotal/MO2)

Similarly, for N2, we have:

MN2 = 28.02 g/mol

mN2/V = dN2 = xN2n/V(MN2)

PN2 = xN2Ptotal = (mN2/V)(RT/MN2) = dN2RT(Ptotal/MN2)

Substituting the values of Ptotal, R, T, MO2, and MN2, we get:

PO2 = 0.231 atm

PN2 = 1.269 atm

Then, the mole fraction of O2 is:

xO2 = PO2/Ptotal = 0.154

And the mole fraction of N2 is:

xN2 = PN2/Ptotal = 0.846

Finally, we can calculate the molar density:

n/V = P/(RT) = Ptotal/(RT)(xO2 + xN2)

Substituting the values, we get:

n/V = (1.50 atm)/(0.0821 L·atm/mol·K)(298 K)(0.154 + 0.846) = 61.3 mol/m³

Therefore, the answer is B.

An Ideal Gas Composed of a Mixture of Oxygen and Nitrogen