Problem 84 - Chemistry The Central Science

Question

Suppose you have two 1 -L flasks, one containing \(\mathrm{N}_{2}\) at STP, the other containing \(\mathrm{CH}_{4}\) at STP. How do these systems compare with respect to (a) number of molecules, (b) density, (c) average kinetic energy of the molecules, (d) rate of effusion through a pinhole leak?

Step-by-Step Solution

Answer

(a) Both N2 and CH4 have the same number of molecules: 2.69 x 10^22 molecules. (b) The density of N2 is 1.25 g/L, and the density of CH4 is 0.713 g/L. (c) The average kinetic energy of the molecules for both N2 and CH4 is 5.655 x 10^-21 J. (d) The ratio of the rate of effusion of N2 to CH4 is approximately \(\sqrt{0.571}\). Thus, N2 effuses at a slower rate than CH4.

Step 1: Calculate number of molecules for both gases at STP

The Ideal Gas law is PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is the temperature. At STP, P = 1 atm and T = 273 K. The volume of each flask is 1 L. n=N2 = (P * V) / (R * T) = (1 atm * 1 L) / (0.08206 L*atm/(K*mol) * 273 K) = 0.0446 mol N2 n=CH4 = (P * V) / (R * T) = (1 atm * 1 L) / (0.08206 L*atm/(K*mol) * 273 K) = 0.0446 mol CH4 Since both gases are in 1 L flasks at STP and follow the Ideal Gas Law, they have the same number of moles. Now we can use Avogadro's number (6.022 x 10^23 molecules/mol) to find the number of molecules for each gas: N(N2) = 0.0446 mol * 6.022 x 10^23 molecules/mol = 2.69 x 10^22 molecules N(CH4) = 0.0446 mol * 6.022 x 10^23 molecules/mol = 2.69 x 10^22 molecules a) N2 and CH4 have the same number of molecules: 2.69 x 10^22 molecules.

Step 2: Calculate the density of both gases

To calculate the density, we need to divide the mass by volume (density = mass/volume). First, we need to find the mass of both gases. The molar masses of N2 and CH4 are: Molar mass of N2: 28 g/mol Molar mass of CH4: 16 g/mol Now, we can find the mass of each gas by multiplying the number of moles by their respective molar masses: Mass of N2: 0.0446 mol * 28 g/mol = 1.25 g Mass of CH4: 0.0446 mol * 16 g/mol = 0.713 g Density of N2: 1.25 g / 1 L = 1.25 g/L Density of CH4: 0.713 g / 1 L = 0.713 g/L b) The density of N2 is 1.25 g/L, and the density of CH4 is 0.713 g/L.

Step 3: Calculate the average kinetic energy of the molecules

The average kinetic energy of the molecules is given by KE_avg = (3/2)kT, where k is Boltzmann's constant (1.381 x 10^-23 J/K) and T is the temperature (in this case, STP = 273 K). Since both gases are at the same temperature, they will have the same average kinetic energy. KE_avg = (3/2)(1.381 x 10^-23 J/K)(273 K) = 5.655 x 10^-21 J c) The average kinetic energy of the molecules for both N2 and CH4 is 5.655 x 10^-21 J.

Step 4: Calculate the rate of effusion

To calculate the rate of effusion, we will use Graham's law, which states that the ratio of the rates of effusion of two gases is inversely proportional to the square root of their molar masses: Rate_A/Rate_B = sqrt(M_B/M_A) We'll use this equation to find the ratio of the rate of effusion of N2 (A) to CH4 (B): Rate_N2/Rate_CH4 = sqrt(M_CH4/M_N2) = sqrt(16 g/mol / 28 g/mol) = sqrt(0.571) d) The ratio of the rate of effusion of N2 to CH4 is approximately sqrt(0.571). Thus, N2 effuses at a slower rate than CH4.