Two moles of an ideal gas with a volume of 4.00 liters and a temperature of $0^{\circ} \mathrm{C}$ are mixed with one mole of an ideal gas with a volume of 3.00 liters and a temperature of $100^{\circ} \mathrm{C}$. The volume of the mixture is 5.00 liters. What is the number density of the mixture?
A) $3.61 \times 10^{26} \mathrm{~m}^{-3}$
B) $1.98 \times 10^{26} \mathrm{~m}^{-3}$
C) $2.02 \times 10^{26} \mathrm{~m}^{-3}$
D) $4.00 \times 10^{26} \mathrm{~m}^{-3}$
E) $2.56 \times 10^{26} \mathrm{~m}^{-3}$

Question

Quizplus · Accepted Answer

The number density of a gas is the number of molecules per unit volume. The number of molecules in a gas is proportional to the number of moles of gas, and the volume of a gas is proportional to the temperature of the gas. Therefore, the number density of a gas is proportional to the number of moles of gas and inversely proportional to the temperature of the gas.In this problem, we have two moles of gas with a volume of 4.00 liters and a temperature of $0^{\circ} \mathrm{C}$, and one mole of gas with a volume of 3.00 liters and a temperature of $100^{\circ} \mathrm{C}$. The volume of the mixture is 5.00 liters. The number of moles of gas in the mixture is 2 + 1 = 3 moles. The temperature of the mixture is the average of the two temperatures, which is $(0^{\circ} \mathrm{C} + 100^{\circ} \mathrm{C}) / 2 = 50^{\circ} \mathrm{C}$. Therefore, the number density of the mixture is:$$n = \frac{3 \mathrm{~mol}}{5.00 \mathrm{~L}} \times \frac{273 \mathrm{~K}}{323 \mathrm{~K}} = 3.61 \times 10^{26} \mathrm{~m}^{-3}$$Therefore, the correct answer is A.

Two Moles of an Ideal Gas with a Volume of 4.00