Question

Consider two bulbs \(\mathrm{A}\) and \(\mathrm{B}\), identical in volume and temperature. Bulb A contains \(1.0 \mathrm{~mol}\) of \(\mathrm{N}_{2}\) and bulb \(\mathrm{B}\) has \(1.0 \mathrm{~mol}\) of \(\mathrm{NH}_{3} .\) Both bulbs are connected by a tube with a valve that is closed. (a) Which bulb has the higher pressure? (b) Which bulb has the gas with the higher density? (c) Which bulb has molecules with a higher average kinetic energy? (d) Which bulb has a gas whose molecules move with a faster molecular speed? (e) If the valve between the two bulbs is opened, how will the pressure change? (f) If \(2.0 \mathrm{~mol}\) of He are added while the valve is opened, what fraction of the total pressure will be due to helium?

Short answer

The pressure in both bulbs A and B is the same, as both bulbs have the same volume, temperature, and number of moles. (b) Which bulb contains a gas with a higher density? Bulb A contains a gas with a higher density, as the mass of nitrogen gas (\(\mathrm{N}_2\)) is greater than ammonia (\(\mathrm{NH}_3\)). (c) What is the comparison of the average kinetic energy of the gas molecules in both bulbs? The average kinetic energy of the gas molecules in both bulbs is the same, as they are at the same temperature. (d) Which bulb contains gas molecules with a higher molecular speed? Bulb B contains gas molecules with a higher molecular speed, as ammonia (\(\mathrm{NH}_3\)) has a lower molar mass compared to nitrogen gas (\(\mathrm{N}_2\)). (e) What is the effect of opening the valve between the two bulbs on the pressure in the system? The effect of opening the valve between the two bulbs is that there is no overall pressure change in the system, as both bulbs have the same pressure and temperature. (f) What is the effect of adding helium to the system on the total pressure? The effect of adding helium to the system is that helium will exert half (50%) of the total pressure, as it contributes 2.0 moles out of a total of 4.0 moles in the system.

Step-by-step solution

(a) Comparing Pressure in Both Bulbs

According to the Ideal Gas Law, \(PV = nRT\). Since both bulbs are identical in volume (V) and temperature (T), the number of moles (n) and the gas constant (R) are the same. So, the pressure (P) in both bulbs is also the same.

(b) Comparing Gas Density in Both Bulbs

To find the density of each gas, we use the formula \(\rho = \frac{m}{V}\), where \(\rho\) is density, m is mass, and V is volume. Since both bulbs have the same volume, we can compare the mass of the gas molecules to determine their densities. The molar mass of \(\mathrm{N}_2\) is \(28.0 \mathrm{~g/mol}\), and for \(\mathrm{NH}_3\), it is \(17.0 \mathrm{~g/mol}\). Since there is \(1.0 \mathrm{~mol}\) of each gas, the mass of \(\mathrm{N}_2\) is greater, thus bulb A has a gas with higher density.

(c) Comparing Average Kinetic Energy in Both Bulbs

The average kinetic energy (\(KE_{avg}\)) of the molecules in an ideal gas is directly proportional to its temperature, as shown by the equation \(KE_{avg} = \frac{3}{2} kT\), where k is Boltzmann's constant and T is the temperature. Since both bulbs are at the same temperature, their molecules have the same average kinetic energy.

(d) Comparing Molecular Speed in Both Bulbs

The molecular speed of a gas can be found using the formula \(v_{rms} = \sqrt{\frac{3RT}{M}}\), where \(v_{rms}\) is the root-mean-square speed, R is the gas constant, T is the temperature, and M is the molar mass of the gas. Since both bulbs are at the same temperature, the molecular speed can be compared based on their molar mass. Bulb A contains \(\mathrm{N}_2\) with a higher molar mass, so it has a slower molecular speed than the \(\mathrm{NH}_3\) in Bulb B.

(e) Effect of Opening Valve on Pressure

When the valve between the two bulbs is opened, the gas molecules will diffuse from their respective bulbs and mix. Since both bulbs have the same pressure and temperature, there will be no overall pressure change once the valve is opened.

(f) Effect of Adding Helium on Total Pressure

If \(2.0 \mathrm{~mol}\) of helium (\(\mathrm{He}\)) is added to the system while the valve is open, we now have three gases to consider. The total pressure, \(P_{total}\), is the sum of the individual pressures exerted by each gas. Assuming the volume and temperature remain constant, the pressure exerted by each gas is proportional to the number of moles present. The total moles of the system become: \(1.0 \mathrm{~mol} \mathrm{(N_2)} + 1.0 \mathrm{~mol} \mathrm{(NH_3)} + 2.0 \mathrm{~mol} \mathrm{(He)} = 4.0 \mathrm{~mol}\). The fraction of total pressure due to helium is: \(\frac{2.0 \mathrm{~mol}}{4.0 \mathrm{~mol}}= \frac{1}{2}\). Hence, helium will exert half (50%) of the total pressure in the system.