Question

Calculate the root mean square velocities of \(\mathrm{CH}_{4}(g)\) and \(\mathrm{N}_{2}(g)\) molecules at 273 \(\mathrm{K}\) and 546 \(\mathrm{K} .\)

Short answer

The root mean square velocities for CH4 and N2 at the given temperatures are: For CH4 at 273 K: \(v_{rms} \approx 922.65\,\mathrm{m/s}\) For CH4 at 546 K: \(v_{rms} \approx 1304.97\,\mathrm{m/s}\) For N2 at 273 K: \(v_{rms} \approx 661.54\,\mathrm{m/s}\) For N2 at 546 K: \(v_{rms} \approx 935.68\,\mathrm{m/s}\)

Step-by-step solution

Write down the equation for the root mean square (rms) velocity

To find the root mean square velocity of a gas, we will use the following formula: \[v_{rms} = \sqrt{\frac{3RT}{M}}\] Where \(v_{rms}\) is the root mean square velocity, \(R\) is the ideal gas constant (8.314 J/mol K), \(T\) is the temperature in Kelvin, and \(M\) is the molecular mass in kg/mol.

Calculate the molecular mass of CH4 and N2

To find the molecular mass of CH4 and N2 in kg/mol, we'll need to multiply the atomic mass of each element by its frequency in the molecule and then convert the result to kg. For CH4: \(M_{CH4} = (12.01\,\mathrm{g/mol}\times 1) + (1.01\,\mathrm{g/mol}\times 4)\) \(M_{CH4} = 12.01\,\mathrm{g/mol} + 4.04\,\mathrm{g/mol}\) \(M_{CH4} = 16.05\,\mathrm{g/mol}\) Converting to kg/mol, we get: \(M_{CH4} = 0.01605\,\mathrm{kg/mol}\) For N2: \(M_{N2}= (14.01\,\mathrm{g/mol}\times 2)\) \(M_{N2} = 28.02\,\mathrm{g/mol}\) Converting to kg/mol, we get: \(M_{N2} = 0.02802\,\mathrm{kg/mol}\)

Calculate the root mean square velocities

Now that we have the molecular masses and the temperature values, we can calculate the root mean square velocities for CH4 and N2 at 273 K and 546 K. For CH4 at 273 K: \(v_{rms} = \sqrt{\frac{3\times 8.314\,\mathrm{J/(mol\cdot K)}\times 273\,\mathrm{K}}{0.01605\,\mathrm{kg/mol}}}\) \(v_{rms} \approx 922.65\,\mathrm{m/s}\) For CH4 at 546 K: \(v_{rms} = \sqrt{\frac{3\times 8.314\,\mathrm{J/(mol\cdot K)}\times 546\,\mathrm{K}}{0.01605\,\mathrm{kg/mol}}}\) \(v_{rms} \approx 1304.97\,\mathrm{m/s}\) For N2 at 273 K: \(v_{rms} = \sqrt{\frac{3\times 8.314\,\mathrm{J/(mol\cdot K)}\times 273\,\mathrm{K}}{0.02802\,\mathrm{kg/mol}}}\) \(v_{rms} \approx 661.54\,\mathrm{m/s}\) For N2 at 546 K: \(v_{rms} = \sqrt{\frac{3\times 8.314\,\mathrm{J/(mol\cdot K)}\times 546\,\mathrm{K}}{0.02802\,\mathrm{kg/mol}}}\) \(v_{rms} \approx 935.68\,\mathrm{m/s}\)

Present the final results

The root mean square velocities for each gas at the given temperatures are: For CH4 at 273 K: \(v_{rms} \approx 922.65\,\mathrm{m/s}\) For CH4 at 546 K: \(v_{rms} \approx 1304.97\,\mathrm{m/s}\) For N2 at 273 K: \(v_{rms} \approx 661.54\,\mathrm{m/s}\) For N2 at 546 K: \(v_{rms} \approx 935.68\,\mathrm{m/s}\)

Key concepts

Kinetic Molecular Theory

The Kinetic Molecular Theory is a model to explain the behavior of gases. It helps us understand how gas molecules move and interact. This theory applies a few basic assumptions:

• Gas consists of a large number of tiny particles (atoms or molecules) that are in constant, random motion.
• These particles collide with each other and the walls of their container. The collisions are perfectly elastic, meaning no energy is lost.
• The average kinetic energy of gas particles depends on the temperature of the gas. Higher temperatures mean more energy and faster moving particles.

From this theory, we can deduce that the root mean square (rms) velocity of gas molecules depends on both the temperature and the molecular mass of the gas. The formula for rms velocity, \[v_{rms} = \sqrt{\frac{3RT}{M}}\], directly ties the motion of particles to these factors. Here, \(R\) is the ideal gas constant, \(T\) is the temperature, and \(M\) is the molecular mass. This formula shows why lighter molecules or greater temperatures lead to higher rms velocities.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that mathematically describes the state of an ideal gas. The equation is written as:\[PV = nRT\]where \(P\) represents pressure, \(V\) is volume, \(n\) is the number of moles of the gas, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin. This law implies that for a given amount of gas at a constant temperature, if the pressure increases, the volume must decrease, and vice versa. This is known as Boyle's law. Similarly, if the temperature increases, either the volume or pressure must increase if the other variable is held constant. This is Charles's law.The Ideal Gas Law is important when calculating the root mean square velocity since it utilizes the relationship between the temperature and pressure of gas to define the state of gas particles more accurately. These principles allow us to predict and manipulate gas behavior in various conditions, crucial for applications in industries and scientific experiments.

Temperature Dependence of Gases

The behavior of gas molecules is significantly influenced by temperature. As temperature increases, the kinetic energy of gas particles also increases. This means that gas particles move faster when they are heated. Conversely, as the temperature decreases, the same particles slow down.

• At higher temperatures, gases expand because the particles move further apart, needing more space to move.
• At low temperatures, gas volume contracts as the motion of the molecules becomes less vigorous.
• The root mean square velocity (\(v_{rms}\)) of gas molecules is proportional to the square root of the temperature (\(T\)). Hence, \(v_{rms} \propto \sqrt{T}\).

This relationship shows why the rms velocity at 546 K is higher than at 273 K. As explained in the concept of kinetic molecular theory, the speed of molecules increases with temperature, confirming that higher temperatures make molecules move faster.