Problem 68 - College Physics

Question

For an average molecule of \(\mathrm{N}_{2}\) gas at \(10^{\circ} \mathrm{C}\), what are its (a) translational kinetic energy, (b) rotational kinetic energy, and (c) total energy? Repeat for He gas at the same temperature.

Step-by-Step Solution

Answer

For \(N_2\), KE_trans = \(5.865 \times 10^{-21}\) J, KE_rot = \(3.906 \times 10^{-21}\) J, total energy = \(9.771 \times 10^{-21}\) J. For He, total energy = \(5.865 \times 10^{-21}\) J.

Step 1: Understand Concepts

The exercise involves finding the energies of an average molecule in nitrogen ( N_2) and helium (He) gases at a specific temperature. We'll calculate translational kinetic energy, rotational kinetic energy for N_2 (since He is a monatomic gas and doesn't rotate), and total energy for both gases.

Step 2: Identify Formulas

For a molecule in an ideal gas:- Translational kinetic energy (for both gases) is given by: \[ KE_{trans} = \frac{3}{2} k_B T \]- Rotational kinetic energy for diatomic gases like N_2: \[ KE_{rot} = k_B T \]- Total energy is the sum for N_2: \[ E_{total_{N_2}} = KE_{trans} + KE_{rot} \]For helium (He, monatomic), rotational energy is 0, and thus: \[ E_{total_{He}} = KE_{trans} \]

Step 3: Convert Temperature

Convert the temperature from Celsius to Kelvin: \[ T(K) = T(^\circ C) + 273.15 = 10 + 273.15 = 283.15 \, K \]

Step 4: Calculate Translational Kinetic Energy

Use the formula for translational kinetic energy: \[ KE_{trans} = \frac{3}{2} k_B T \]Where:- \( k_B = 1.38 \times 10^{-23} \; \text{J/K} \) is the Boltzmann constant- \( T = 283.15 \; \text{K} \)Substitute and solve: \[ KE_{trans} = \frac{3}{2} \times 1.38 \times 10^{-23} \times 283.15 = 5.865 \times 10^{-21} \; \text{J} \]

Step 5: Calculate Rotational Kinetic Energy for \(N_2\)

Use the formula for rotational kinetic energy for diatomic gases: \[ KE_{rot} = k_B T \]Substitute and solve: \[ KE_{rot} = 1.38 \times 10^{-23} \times 283.15 = 3.906 \times 10^{-21} \; \text{J} \]

Step 6: Calculate Total Energy for \(N_2\)

Add translational and rotational kinetic energies for \(N_2\): \[ E_{total_{N_2}} = KE_{trans} + KE_{rot} = 5.865 \times 10^{-21} + 3.906 \times 10^{-21} = 9.771 \times 10^{-21} \; \text{J} \]

Step 7: Calculate Total Energy for \(He\)

Since helium is monatomic, the total energy is just translational kinetic energy: \[ E_{total_{He}} = KE_{trans} = 5.865 \times 10^{-21} \; \text{J} \]

Key Concepts

Translational Kinetic Energy

Translational kinetic energy is the energy possessed by a molecule as it moves from one place to another. It’s a type of kinetic energy that applies to all molecules in gases, whether they are diatomic, like nitrogen (2), or monatomic, like helium (He).
To determine translational kinetic energy in an ideal gas, use the formula:

• \( KE_{trans} = \frac{3}{2} k_B T \)

Here, \( k_B \) is the Boltzmann constant \((1.38 \times 10^{-23} \, \text{J/K})\), and \( T \) is the temperature in Kelvin.
The equation gives us a straightforward way to calculate how temperature affects the motion of gas molecules. Higher temperatures increase the kinetic energy due to greater molecular motion.

Rotational Kinetic Energy

Rotational kinetic energy comes into play for molecules that can rotate, such as diatomic gases like nitrogen (2). It accounts for the energy due to the rotation of molecules.
To calculate rotational kinetic energy, we use the formula:

• \( KE_{rot} = k_B T \)

Since helium (He) atoms are monatomic, they do not have rotational kinetic energy; they don't rotate like diatomic molecules do.
This concept is crucial in understanding the energy balance in the molecules of a gas, where rotational modes also contribute to the overall energy besides translational motion.

Ideal Gas

An ideal gas is a theoretical model used to understand the behavior of gases. It's helpful because the properties of an ideal gas closely resemble those of real gases under many conditions, simplifying calculations.
Key assumptions of an ideal gas include:

• Gas molecules have no volume.
• No intermolecular forces exist between the gas molecules.
• Molecules undergo perfectly elastic collisions.

These assumptions allow us to use simple formulas to express the energy contained in gases. Moreover, the behavior of gases described with kinetic energy calculations as shown above primarily uses ideal gas approximations, making it an essential concept in gas kinetics.