Problem 80 - Chemistry The Central Science In Si Units

Question

Suppose you have two 1 -L flasks, one containing \(\mathrm{N}_{2}\) at \(\mathrm{STP}\), 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, \((\mathbf{d})\) rate of effusion through a pinhole leak?

Step-by-Step Solution

Answer

(a) Equal number of molecules, (b) \( \mathrm{N}_2 \) is denser, (c) Same average kinetic energy, (d) \( \mathrm{CH}_4 \) effuses faster.

Step 1: Understanding STP Conditions

Standard Temperature and Pressure (STP) are defined as a temperature of 273.15 K (0°C) and a pressure of 1 atm. Under these conditions, 1 mole of an ideal gas occupies 22.4 L.

Step 2: Determining Number of Molecules

At STP, each 1-L flask contains \( \frac{1}{22.4} \) moles of the gas. Since 1 mole contains Avogadro's number of molecules \(6.022 \times 10^{23}\), each flask will contain \( \frac{6.022 \times 10^{23}}{22.4} \) molecules. As both are the same under STP, each flask has the same number of molecules.

Step 3: Calculating Density

Density is mass divided by volume. Molar mass of \( \mathrm{N}_2 \) is 28 g/mol and \( \mathrm{CH}_4 \) is 16 g/mol. At STP, each gas occupies 22.4 L per mole in the flask, so densities are: \( \frac{28}{22.4} \) g/L for \( \mathrm{N}_2 \) and \( \frac{16}{22.4} \) g/L for \( \mathrm{CH}_4 \). Therefore, \( \mathrm{N}_2 \) has a higher density.

Step 4: Average Kinetic Energy of Molecules

Kinetic energy of a gas depends on temperature. Since both gases are at STP, and thus the same temperature, the average kinetic energy of molecules in both gases is the same, determined by \( \frac{3}{2} kT \), where \( k \) is the Boltzmann constant and \( T \) is temperature.

Step 5: Rate of Effusion

Rate of effusion is inversely proportional to the square root of the molar mass of the gas \( \left( R \propto \frac{1}{\sqrt{M}} \right) \). The molar mass of \( \mathrm{N}_2 \) is 28 g/mol and \( \mathrm{CH}_4 \) is 16 g/mol. Consequently, \( \mathrm{CH}_4 \) will effuse faster than \( \mathrm{N}_2 \).

Key Concepts

Molecular Density Comparison

When examining gases in containers, molecular density becomes a key property to compare. Density is calculated by dividing the mass of the gas by its volume. For gases like \( \mathrm{N}_2 \) and \( \mathrm{CH}_4 \) at Standard Temperature and Pressure (STP), their molar mass plays an essential role in determining density. - The molar mass of \( \mathrm{N}_2 \) is 28 g/mol and for \( \mathrm{CH}_4 \), it's 16 g/mol.- Both gases at STP conditions occupy a volume of 22.4 L per mole.Thus, to find the density:- \( \mathrm{N}_2 \) density is \( \frac{28 \text{ g/mol}}{22.4 \text{ L/mol}} \approx 1.25 \text{ g/L} \)- \( \mathrm{CH}_4 \) density is \( \frac{16 \text{ g/mol}}{22.4 \text{ L/mol}} \approx 0.71 \text{ g/L} \)

Kinetic Energy of Gases

Kinetic energy in gases is a fascinating topic because it provides insights into gas behavior. This energy depends primarily on the temperature of the gas. At STP, both \( \mathrm{N}_2 \) and \( \mathrm{CH}_4 \) are at the same temperature—273.15 K.Due to this:- The average kinetic energy for any ideal gas is given by \( \frac{3}{2} kT \), where \( k \) is the Boltzmann constant and \( T \) is the gas temperature.- Thus, at STP, \( \mathrm{N}_2 \) and \( \mathrm{CH}_4 \) both have equal kinetic energies because they share the same temperature.

Rate of Effusion

Effusion is the process by which gas escapes through a tiny hole into a vacuum. The rate at which this occurs is fascinating as it varies depending on the gas type. Graham's Law explains that the rate of effusion is inversely proportional to the square root of the molar mass of the gas. Simply put:\[ R \propto \frac{1}{\sqrt{M}} \]where \( R \) is the rate of effusion and \( M \) is the molar mass.- For \( \mathrm{N}_2 \), \( M = 28 \text{ g/mol} \)- For \( \mathrm{CH}_4 \), \( M = 16 \text{ g/mol} \)

STP Conditions in Chemistry

Standard Temperature and Pressure (STP) is a set of conditions used in chemistry to facilitate comparisons between different gas properties. At STP: - The temperature is defined at 273.15 K (0°C). - The pressure is set at 1 atm. Under these conditions, one mole of an ideal gas occupies a volume of exactly 22.4 L. The consistency of STP allows scientists and students to compare gas behaviors without variables like pressure and temperature fluctuations. These fixed conditions: - Make calculations straightforward when determining properties like density, kinetic energy, and effusion rates. - Provide a baseline to focus purely on the nature of the gas itself rather than how external factors alter its behavior.