Question

Consider a sample of ideal gas molecules for the following question. a. How is the average kinetic energy of the gas molecules related to temperature? b. How is the average velocity of the gas molecules related to temperature? c. How is the average velocity of the gas molecules related to the molar mass of the gas at constant temperature?

Short answer

a. The average kinetic energy of the gas molecules is directly proportional to the temperature, given by the equation \(K.E._{avg} = \frac{3}{2} kT\). b. The magnitude of the average velocity of the gas molecules is related to the square root of the temperature, as \(v_{rms} \propto \sqrt{T}\). c. At constant temperature, the magnitude of the average velocity of the gas molecules is inversely proportional to the square root of the molar mass of the gas, given by the equation \(v_{rms} \propto \frac{1}{\sqrt{M}}\).

Step-by-step solution

a. Relationship between average kinetic energy and temperature

The average kinetic energy of a gas molecule is given by the equipartition theorem, which states that for an ideal gas, the average kinetic energy per degree of freedom per molecule is proportional to the temperature of the gas. Mathematically, this is expressed as: \(K.E._{avg} = \frac{3}{2} kT\) where \(K.E._{avg}\) is the average kinetic energy, \(k\) is the Boltzmann constant, and \(T\) is the temperature of the gas in Kelvin. Thus, the average kinetic energy of the gas molecules is directly proportional to the temperature.

b. Relationship between average velocity and temperature

In order to find the relationship between the average velocity of the gas molecules and the temperature, we first need to find the root mean square (RMS) velocity, which is defined as: \(v_{rms} = \sqrt{\frac{3kT}{m}}\) where \(v_{rms}\) is the root mean square velocity, \(m\) is the mass of a single gas molecule, and \(k\) and \(T\) are as defined earlier. The RMS velocity, however, is not a direct measure of the average velocity of the gas molecules. For an ideal gas, the average velocity is zero due to random motion of the gas molecules in all directions. But, the RMS velocity tells us about the magnitude of the velocities. The RMS velocity is directly related to the temperature as it obeys the same relationship as found earlier: \(v_{rms} \propto \sqrt{T}\) Thus, the magnitude of the average velocity of the gas molecules is related to the square root of the temperature.

c. Relationship between average velocity and molar mass at constant temperature

To find the relationship between average velocity and molar mass at constant temperature, we can rewrite the equation for the RMS velocity in terms of the molar mass. Instead of using the mass of a single molecule, we can use the molar mass, \(M\), which is the mass of one mole of gas molecules: \(v_{rms} = \sqrt{\frac{3kT}{(M/N_A)}}\) where \(N_A\) is Avogadro's number (approximately \(6.022 \times 10^{23}\)) and \(M\) is the molar mass in kg/mol. We can now rewrite this equation in terms of the molar gas constant, \(R\), which is given by: \(R = kN_A\) Replacing \(kN_A\) with \(R\) in our previous equation, we get: \(v_{rms} = \sqrt{\frac{3RT}{M}}\) At constant temperature, the term \(\frac{3R}{M}\) is constant. Therefore, the RMS velocity is inversely proportional to the square root of the molar mass: \(v_{rms} \propto \frac{1}{\sqrt{M}}\) In conclusion, at constant temperature, the magnitude of the average velocity of the gas molecules is inversely proportional to the square root of the molar mass of the gas.

Key concepts

Average Kinetic Energy

In the world of ideal gas molecules, the concept of average kinetic energy is fundamental. The average kinetic energy of gas molecules is a measure of how much energy each molecule has due to its motion. The beauty of this concept lies in its direct relationship with the temperature of the gas.
The formula that ties these two together is given by:\[K.E._{avg} = \frac{3}{2} kT\]Here, \(K.E._{avg}\) represents the average kinetic energy, \(k\) is the Boltzmann constant, a tiny number that allows us to link temperature with energy. Lastly, \(T\) is the temperature, measured in Kelvin. This formula tells us that as the temperature increases, the kinetic energy of the gas molecules also increases.
The fact that kinetic energy is proportional to temperature helps us understand how heating a gas speeds up the molecules. It's why a hot air balloon rises, as warming the air inside increases the energy and therefore the speed of the molecules, causing the balloon to inflate and lift.

Temperature

Temperature is more than just a number on a thermometer; it's a crucial player in understanding the behavior of gases. The temperature of a gas provides a way to assess how fast and how energetically gas molecules move. When discussing gas properties, especially in the context of ideal gases, temperature is expressed in Kelvin.
To see how temperature affects gas behavior, consider the root mean square velocity (RMS velocity). This is a statistical measure of the speed of particles in a gas, given by the equation:\[v_{rms} = \sqrt{\frac{3kT}{m}}\]Though it's not exactly an average, it gives insight into the typical speed of particles. As the temperature \(T\) increases, \(v_{rms}\) also increases, indicating faster particle movement. This is why heating up a gas causes it to expand, as the particles move quickly, taking up more space.
Even though the average velocity of gas molecules may technically be zero due to their random motion in different directions, the RMS velocity helps us understand how temperature enhances their kinetic activity.

Molar Mass

Molar mass is a critical factor in determining how gas molecules move at a given temperature. The molar mass of a gas is the mass of one mole of its molecules, and it plays a pivotal role in understanding the relationship between temperature and molecular velocity.
The root mean square velocity equation can be rewritten in terms of molar mass \(M\):\[v_{rms} = \sqrt{\frac{3RT}{M}}\]In this version, \(R\) is the molar gas constant, and \(T\) is temperature in Kelvin. This equation reveals that the RMS velocity is inversely proportional to the square root of the molar mass. This means that lighter gases have higher velocities compared to heavier gases at the same temperature.
Intuitively, if two gases are at the same temperature, the one with the lower molar mass will have particles moving faster due to their reduced mass. This principle is why helium balloons float higher and faster than air-filled ones; helium, being lighter, needs less energy per molecule to move quickly.