Internal Molar Energy of Linear Molecule Calculator

Molar Internal Energy Formula:

\[ U_{molar} = \frac{3}{2}RT + \left(\frac{1}{2}I_y\omega_y^2 + \frac{1}{2}I_z\omega_z^2\right) + (3N-5)RT \]

Temperature (T):

Moment of Inertia (Y-axis, I_y):

kg·m²

Angular Velocity (Y-axis, ω_y):

rad/s

Moment of Inertia (Z-axis, I_z):

kg·m²

Angular Velocity (Z-axis, ω_z):

rad/s

Atomicity (N):

Molar Internal Energy (U_molar):

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1. What is Molar Internal Energy of a Linear Molecule?

Definition: The molar internal energy of a linear molecule is the total energy contained within one mole of the substance, including translational, rotational, and vibrational contributions.

Purpose: This calculation is essential in thermodynamics and statistical mechanics for understanding energy distribution in molecular systems.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ U_{molar} = \frac{3}{2}RT + \left(\frac{1}{2}I_y\omega_y^2 + \frac{1}{2}I_z\omega_z^2\right) + (3N-5)RT \]

Where:

\( U_{molar} \) — Molar internal energy (J/mol)
\( R \) — Universal gas constant (8.314 J/mol·K)
\( T \) — Temperature (K)
\( I_y, I_z \) — Moments of inertia about Y and Z axes (kg·m²)
\( \omega_y, \omega_z \) — Angular velocities about Y and Z axes (rad/s)
\( N \) — Atomicity (number of atoms in molecule)

Explanation: The formula combines translational (\(\frac{3}{2}RT\)), rotational (\(\frac{1}{2}I\omega^2\)), and vibrational (\((3N-5)RT\)) energy contributions.

3. Importance of Molar Internal Energy Calculation

Details: Accurate calculation of internal energy is crucial for predicting thermodynamic properties, reaction kinetics, and phase behavior of molecular systems.

4. Using the Calculator

Tips: Enter all required parameters with appropriate units. For diatomic molecules (N=2), the vibrational term becomes (3*2-5)RT = RT.

5. Frequently Asked Questions (FAQ)

Q1: Why are there two rotational terms?
A: Linear molecules have two independent axes of rotation (Y and Z), each contributing to the rotational energy.

Q2: What if my molecule is non-linear?
A: For non-linear molecules, the formula changes to account for three rotational degrees of freedom.

Q3: What's the physical meaning of (3N-5)?
A: For linear molecules, this represents the number of vibrational degrees of freedom (3N-5), where N is atomicity.

Q4: How do I determine moments of inertia?
A: Moments of inertia can be calculated from molecular geometry or measured experimentally.

Q5: Does this include electronic energy?
A: No, this formula only accounts for translational, rotational, and vibrational energies at normal temperatures.