Question

1. consider two gases, a and b, each in a 1.0 - l container with both gases at the same temperature and pressure. the mass of gas a in the container is 0.34 g and the mass of gas b in the container is 0.48 g. a. which gas sample has the most molecules present? explain. b. which gas sample has the largest average kinetic energy? explain. c. which gas sample has the fastest average velocity? explain. d. how can the pressure in the two containers be equal to each other since the larger gas b molecules collide with the container walls more forcefully?

Explanation:

Step1: Apply Avogadro's law

At the same temperature, pressure and volume, the number of moles of a gas is proportional to the number of molecules. Since $n=\frac{m}{M}$ (where $n$ is the number of moles, $m$ is the mass and $M$ is the molar - mass), and from Avogadro's law $n_1/n_2 = N_1/N_2$ (where $N$ is the number of molecules). For gases A and B at the same $T$, $P$ and $V$, the number of moles is directly related to mass. Gas A has a mass of $m_A = 0.34\ g$ and gas B has a mass of $m_B=0.48\ g$. Since $m_B>m_A$, gas B has more moles and thus more molecules.

Step2: Recall the relationship between average kinetic energy and temperature

The average kinetic energy of gas molecules is given by $\overline{KE}=\frac{3}{2}kT$, where $k$ is the Boltzmann constant and $T$ is the temperature. Since both gases are at the same temperature, the average kinetic energy of gas A and gas B is the same.

Step3: Use the root - mean - square speed formula

The root - mean - square speed of gas molecules is $u_{rms}=\sqrt{\frac{3RT}{M}}$. Since $n=\frac{m}{M}$ and at the same $T$, $P$ and $V$, $n$ is proportional to $m$. Gas A has less mass, so it has a lower molar mass (assuming they are pure substances). A lower molar mass $M$ in the formula $u_{rms}=\sqrt{\frac{3RT}{M}}$ means a higher average velocity. So gas A has the fastest average velocity.

Step4: Consider the ideal gas law and molecular properties

The pressure of a gas is given by the ideal gas law $PV = nRT$. The pressure depends on the number of moles $n$ of the gas, the temperature $T$ and the volume $V$. Although gas B molecules collide more forcefully, gas A has more molecules per unit volume (because it has a lower molar mass and more moles at the given mass). The product of the frequency of collisions and the force of collisions is the same for both gases, resulting in equal pressures.

Answer:

a. Gas B has the most molecules present because, at the same temperature, pressure and volume, the number of moles is proportional to the mass, and gas B has a greater mass.
b. Both gas A and gas B have the same average kinetic energy because the average kinetic energy of gas molecules depends only on the temperature, and they are at the same temperature.
c. Gas A has the fastest average velocity. Since at the same $T$, $P$ and $V$, gas A has a lower molar - mass (due to its lower mass for the same number of moles as determined by the conditions), and the average velocity is inversely proportional to the square - root of the molar mass.
d. Although gas B molecules collide more forcefully with the container walls, gas A has a greater number of molecules per unit volume (because of its lower molar mass for the given mass at the same $T$, $P$ and $V$). The product of the frequency of collisions and the force of collisions is the same for both gases, resulting in equal pressures.