The **arrhenius equation calculator** is a computational tool designed to analyze the **temperature dependence** of **chemical reaction rates with k = A × e^(-Ea/RT) formula**.

The calculator implements the fundamental **Arrhenius equation**, proposed by Swedish chemist **Svante Arrhenius** in 1889.

**Consider a reaction where k₁ = 2.33 × 10⁻³ s⁻¹ at T₁ = 300 K, and we need to find k₂ at T₂ = 315 K with Eₐ = 55.5 kJ/mol.**

**Using the calculator:**

- Input known values
- The calculator processes through the
**exponential relationship** - Outputs k₂ = 6.47 × 10⁻³ s⁻¹

## Arrhenius Equation Calculator

Temperature (K) | Rate Constant (s⁻¹) | Activation Energy (kJ/mol) | Pre-exponential Factor (s⁻¹) |
---|---|---|---|

298 | 1.86 × 10⁻⁴ | 50.0 | 5.0 × 10¹⁰ |

308 | 5.24 × 10⁻⁴ | 50.0 | 5.0 × 10¹⁰ |

318 | 1.39 × 10⁻³ | 50.0 | 5.0 × 10¹⁰ |

323 | 2.56 × 10⁻³ | 50.0 | 5.0 × 10¹⁰ |

333 | 4.76 × 10⁻³ | 50.0 | 5.0 × 10¹⁰ |

343 | 8.77 × 10⁻³ | 50.0 | 5.0 × 10¹⁰ |

353 | 1.62 × 10⁻² | 50.0 | 5.0 × 10¹⁰ |

363 | 2.99 × 10⁻² | 50.0 | 5.0 × 10¹⁰ |

373 | 5.56 × 10⁻² | 50.0 | 5.0 × 10¹⁰ |

383 | 1.03 × 10⁻¹ | 50.0 | 5.0 × 10¹⁰ |

393 | 1.92 × 10¹ | 50.0 | 5.0 × 10¹⁰ |

## Arrhenius Equation Formula

The **Arrhenius equation** is mathematically expressed as:

**k = A × e^(-Eₐ/RT)**

Where:

**k**represents the**rate constant****A**is the**pre-exponential factor****Eₐ**is the**activation energy****R**is the universal gas constant (8.314 J/mol·K)**T**is the absolute temperature in Kelvin

**For a reaction with A = 5.0 × 10¹⁰ s⁻¹, Eₐ = 50 kJ/mol at T = 298 K:
k = (5.0 × 10¹⁰) × e^(-50,000/(8.314 × 298)) = 1.86 × 10⁻⁴ s⁻¹**

## What is an Arrhenius equation?

The **Arrhenius equation** represents a fundamental relationship in physical chemistry that describes how the **rate constant** of a chemical reaction depends on **temperature** and **activation energy**. This equation embodies the observation that most chemical reactions proceed faster at higher temperatures.

**Practical Example**:

The decomposition of hydrogen peroxide (H₂O₂) follows Arrhenius behavior:

- At 20°C: k = 1.0 × 10⁻⁴ s⁻¹
- At 30°C: k = 2.7 × 10⁻⁴ s⁻¹

## How is the Arrhenius equation calculated?

The calculation process involves these steps:

**Logarithmic Form**: ln(k) = ln(A) – (Eₐ/R)(1/T)**Two-Point Form**: ln(k₂/k₁) = -(Eₐ/R)(1/T₂ – 1/T₁)

**Given k₁ = 1.0 × 10⁻⁴ s⁻¹ at T₁ = 293 K and k₂ = 2.7 × 10⁻⁴ s⁻¹ at T₂ = 303 K:**

**ln(2.7 × 10⁻⁴/1.0 × 10⁻⁴) = -(Eₐ/8.314)(1/303 - 1/293)**

Solving for Eₐ = 52.6 kJ/mol

## How do you calculate the pre-exponential factor?

The **pre-exponential factor (A)** can be calculated by rearranging the Arrhenius equation:

**A = k/e^(-Eₐ/RT)**

With k = 1.0 × 10⁻⁴ s⁻¹, Eₐ = 52.6 kJ/mol, T = 293 K:

A = (1.0 × 10⁻⁴)/e^(-52,600/(8.314 × 293)) = 4.2 × 10⁸ s⁻¹

## Activation Energy at Two Temperatures

The equation for calculating activation energy using two temperature points is:

**Eₐ = -R × ln(k₂/k₁) × (T₁T₂/(T₂-T₁))**

For a reaction with:

- k₁ = 2.0 × 10⁻² s⁻¹ at T₁ = 300 K
- k₂ = 8.0 × 10⁻² s⁻¹ at T₂ = 320 K

**Eₐ = -(8.314) × ln(8.0 × 10⁻²/2.0 × 10⁻²) × (300 × 320/20) = 46.3 kJ/mol**

## References:

- Physical Chemistry Chemical Physics: https://pubs.rsc.org/en/journals/journalissues/cp
- Chemical Reviews: https://pubs.acs.org/journal/chreay

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