Thermodynamic Calculations

Tools and formulas for thermodynamic analysis of battery electrolytes.

Physical Constants

# Gas constant
R = 8.314  # J/(mol*K)
R_kj = 0.008314  # kJ/(mol*K)

# Faraday constant
F = 96485  # C/mol

# Boltzmann constant
kB = 1.38e-23  # J/K

Arrhenius Equation

The Arrhenius equation describes temperature dependence of ionic conductivity:

import numpy as np

def arrhenius_conductivity(T, A, Ea):
    """
    Calculate ionic conductivity using Arrhenius equation.

    Parameters:
    - T: Temperature in Kelvin
    - A: Pre-exponential factor (S/cm)
    - Ea: Activation energy (J/mol)

    Returns:
    - sigma: Ionic conductivity (S/cm)
    """
    R = 8.314  # J/(mol*K)
    return A * np.exp(-Ea / (R * T))

Activation Energy from Linear Fit

def calculate_activation_energy(slope):
    """
    Calculate activation energy from Arrhenius plot slope.

    For ln(sigma) vs 1/T plot:
    slope = -Ea/R

    Parameters:
    - slope: Slope from linear regression of ln(sigma) vs 1/T

    Returns:
    - Ea_kjmol: Activation energy in kJ/mol
    """
    R = 8.314  # J/(mol*K)
    Ea_jmol = -slope * R
    Ea_kjmol = Ea_jmol / 1000
    return Ea_kjmol

Temperature Conversions

def celsius_to_kelvin(temp_c):
    """Convert Celsius to Kelvin."""
    return temp_c + 273.15

def kelvin_to_celsius(temp_k):
    """Convert Kelvin to Celsius."""
    return temp_k - 273.15

Complete Arrhenius Analysis

from scipy import stats
import numpy as np

def arrhenius_analysis(temp_c, conductivity):
    """
    Perform complete Arrhenius analysis.

    Parameters:
    - temp_c: Array of temperatures in Celsius
    - conductivity: Array of conductivity values (S/cm)

    Returns:
    - dict: Activation energy, pre-exponential factor, R-squared
    """
    R = 8.314  # J/(mol*K)

    # Convert to Arrhenius form
    temp_k = temp_c + 273.15
    inv_t = 1 / temp_k
    ln_sigma = np.log(conductivity)

    # Linear regression
    result = stats.linregress(inv_t, ln_sigma)

    # Calculate parameters
    Ea_kjmol = -result.slope * R / 1000
    A = np.exp(result.intercept)
    r_squared = result.rvalue ** 2

    return {
        'activation_energy_kjmol': Ea_kjmol,
        'pre_exponential_factor': A,
        'r_squared': r_squared
    }

Typical Values for Li-ion Electrolytes

• Activation energy: 10-40 kJ/mol
• Room temperature conductivity: 1-10 mS/cm
• Pre-exponential factor: 10^3 - 10^6 S/cm