Internal Molar Energy of Non-Linear Molecule Calculator

Molar Internal Energy Formula:

\[ U_{molar} = \frac{3}{2}[R]T + \left( \frac{1}{2}I_y\omega_y^2 + \frac{1}{2}I_z\omega_z^2 + \frac{1}{2}I_x\omega_x^2 \right) + (3N-6)[R]T \]

Temperature (T):

Atomicity (N):

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

Moment of Inertia X-axis (I_x):

kg·m²

Angular Velocity X-axis (ω_x):

rad/s

Molar Internal Energy (U_molar):

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

Definition: The total energy contained within a mole of a non-linear molecule, including translational, rotational, and vibrational components.

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

2. How Does the Calculator Work?

The calculator uses the formula:

\[ U_{molar} = \frac{3}{2}[R]T + \left( \frac{1}{2}I_y\omega_y^2 + \frac{1}{2}I_z\omega_z^2 + \frac{1}{2}I_x\omega_x^2 \right) + (3N-6)[R]T \]

Where:

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

Explanation: The formula sums the translational (3/2RT), rotational (½Iω² for each axis), and vibrational ((3N-6)RT) energy contributions.

3. Importance of Molar Internal Energy Calculation

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

4. Using the Calculator

Tips: Enter all required parameters. For non-rotating molecules, set angular velocities to zero. Temperature must be > 0K.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between linear and non-linear molecules?
A: Non-linear molecules have three rotational degrees of freedom (Ix, Iy, Iz) while linear molecules have only two.

Q2: Why is the vibrational term (3N-6)RT?
A: For non-linear molecules, there are 3N-6 vibrational degrees of freedom (3N total minus 3 translational minus 3 rotational).

Q3: What if my molecule isn't rotating?
A: Set all angular velocities (ω) to zero, and the rotational term will be zero.

Q4: How do I determine moments of inertia?
A: Moments of inertia depend on molecular structure and can be calculated from bond lengths and angles or measured experimentally.

Q5: What are typical angular velocity values?
A: For molecules, angular velocities are typically in the range of 10¹¹ to 10¹³ rad/s, depending on temperature.