Total Entropy Change (Universe) Equation, Derivation & Rearrangements

Core idea

Overview

The Second Law of Thermodynamics states that the total entropy of an isolated system can only increase over time, or remain constant in ideal cases. This equation quantifies this fundamental principle by defining the total entropy change of the universe as the sum of the entropy change within a specific system and its immediate surroundings. A positive value for \Delta S_{universe} indicates a spontaneous process, driving towards greater disorder and energy dispersal.

When to use: Use this equation to determine the spontaneity of a process. If \Delta S_{universe} is positive, the process is spontaneous. It's particularly useful in chemical thermodynamics to predict reaction feasibility and understand energy flow.

Why it matters: Understanding total entropy change is crucial for predicting the direction and feasibility of chemical reactions and physical processes. It underpins the concept of spontaneity and equilibrium, guiding the design of efficient chemical processes and understanding natural phenomena from biological systems to cosmological evolution.

Symbols

Variables

$Δ$ $S_{sy s t e m}$ = Entropy Change of System, $Δ$ $S_{s u r r o u n d in g s}$ = Entropy Change of Surroundings, $Δ$ $S_{u ni v er se}$ = Total Entropy Change of Universe

Δ S_{sy s t e m}

Entropy Change of System

J K^{- 1} m o l^{- 1}

Δ S_{s u r r o u n d in g s}

Entropy Change of Surroundings

J K^{- 1} m o l^{- 1}

Δ S_{u ni v er se}

Total Entropy Change of Universe

J K^{- 1} m o l^{- 1}

Walkthrough

Derivation

Formula: Total Entropy Change (Universe)

The total entropy change of the universe is the sum of the entropy changes of the system and its surroundings, reflecting the Second Law of Thermodynamics.

The system and surroundings together constitute the entire universe relevant to the process.
Entropy is a state function, meaning its change depends only on the initial and final states, not the path taken.

1

Define the Universe:

The universe, in thermodynamic terms, is considered an isolated system comprising the system under observation and its surroundings.

Δ S_{u ni v er se} = Δ S_{t o t a l}

2

Partitioning Entropy Change:

The total entropy change is the sum of the entropy change occurring within the system and the entropy change occurring in its surroundings.

Δ S_{u ni v er se} = Δ S_{sy s t e m} + Δ S_{s u r r o u n d in g s}

Note: This fundamental relationship is a direct consequence of the Second Law of Thermodynamics, which states that for any spontaneous process, the total entropy of the universe must increase.

Result

Δ S_{u ni v er se} = Δ S_{sy s t e m} + Δ S_{s u r r o u n d in g s}

Source: AQA A-level Chemistry — Physical Chemistry (3.1.6 Thermodynamics)

Free formulas

Rearrangements

Solve for $Δ S_{sy s t e m}$

Total Entropy Change (Universe): Make $Δ$ $S_{sy s t e m}$ the subject

Δ S_{sy s t e m} = Δ S_{u ni v er se} - Δ S_{s u r r o u n d in g s}

To make $Δ$ $S_{sy s t e m}$ the subject, subtract $Δ$ $S_{s u r r o u n d in g s}$ from both sides of the equation.

Difficulty: 1/5

Solve for $Δ S_{s u r r o u n d in g s}$

Total Entropy Change (Universe): Make $Δ$ $S_{s u r r o u n d in g s}$ the subject

Δ S_{s u r r o u n d in g s} = Δ S_{u ni v er se} - Δ S_{sy s t e m}

To make $Δ$ $S_{s u r r o u n d in g s}$ the subject, subtract $Δ$ $S_{sy s t e m}$ from both sides of the equation.

Difficulty: 1/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Graph unavailable for this formula.

The graph is a straight line with a slope of one, showing that the total entropy change of the universe increases at the same rate as the entropy change of the system. For a chemistry student, large x-values represent processes where the system becomes significantly more disordered, while small x-values indicate a minimal increase or a decrease in system disorder. The most important feature of this linear relationship is that any change in the system's entropy results in an identical numerical change in the total entropy of the universe, provided the surroundings remain constant. The y-intercept specifically represents the contribution of the surroundings to the total entropy change.

Graph type: linear

Why it behaves this way

Intuition

Visualize the universe as divided into a system and its surroundings; the total entropy change is the sum of how disorder changes within each of these two distinct regions.

Δ S_{u ni v er se}

The total change in the degree of disorder or energy dispersal across all matter and energy involved in a process.

This value dictates the spontaneity of a process; a positive value means the process will naturally occur, reflecting the universe's tendency towards greater overall disorder.

Δ S_{sy s t e m}

The change in the degree of disorder or energy dispersal within the specific part of the universe under observation.

It quantifies how much the system itself becomes more or less disordered, or how its internal energy spreads out, independent of its surroundings.

Δ S_{s u r r o u n d in g s}

The change in the degree of disorder or energy dispersal in the environment immediately interacting with the system.

It reflects how the surroundings become more or less disordered, primarily due to heat transfer to or from the system, which spreads or concentrates energy in the environment.

Free study cues

Insight

Canonical usage

All entropy terms (Δ $S_{u}$ niverse, Δ $S_{s}$ ystem, Δ $S_{s}$ urroundings) must be expressed in consistent units, typically Joules per Kelvin (J/K) or Joules per mole-Kelvin (J/(mol·K)) in the SI system.

Common confusion

A common mistake is using inconsistent units for Δ $S_{s}$ ystem and Δ $S_{s}$ urroundings, or mixing total entropy (J/K) with molar entropy (J/(mol·K))

Unit systems

$Δ S$ J/K · This dimension applies to total entropy change. In chemical contexts, entropy changes are often reported per mole of substance or per mole of reaction, in which case the unit becomes J/(mol·K)

One free problem

Practice Problem

A chemical reaction has an entropy change of the system ( $Δ$ $S_{sy s t e m}$ ) of +45 J K⁻¹ mol⁻¹ and an entropy change of the surroundings ( $Δ$ $S_{s u r r o u n d in g s}$ ) of -20 J K⁻¹ mol⁻¹. Calculate the total entropy change of the universe ( $Δ$ $S_{u ni v er se}$ ) for this reaction.

Entropy Change of System45 J K^{-1} mol^{-1}

Entropy Change of Surroundings-20 J K^{-1} mol^{-1}

Solve for: Delta_S_universe

Hint: Remember to sum the two entropy changes.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

When Predicting whether a chemical reaction will occur spontaneously at a given temperature, Total Entropy Change (Universe) is used to calculate Total Entropy Change of Universe from Entropy Change of System and Entropy Change of Surroundings. The result matters because it helps connect measured amounts to reaction yield, concentration, energy change, rate, or equilibrium.

Study smarter

Tips

Ensure consistent units for $Δ$ $S_{sy s t e m}$ and $Δ$ $S_{s u r r o u n d in g s}$ (usually J K⁻¹ mol⁻¹ or J K⁻¹).
A positive $Δ$ $S_{u ni v er se}$ indicates a spontaneous process.
Remember that $Δ$ $S_{s u r r o u n d in g s}$ can often be calculated from $Δ$ $H_{sy s t e m}$ and temperature ( $Δ$ $S_{s u r r o u n d in g s}$ = - $Δ$ $H_{sy s t e m}$ / T).
At equilibrium, $Δ$ $S_{u ni v er se}$ = 0.

Avoid these traps

Common Mistakes

Forgetting to convert units (e.g., kJ to J).
Incorrectly calculating $Δ$ $S_{s u r r o u n d in g s}$ or using the wrong sign.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The total entropy change of the universe is the sum of the entropy changes of the system and its surroundings, reflecting the Second Law of Thermodynamics.

Use this equation to determine the spontaneity of a process. If \Delta S_{universe} is positive, the process is spontaneous. It's particularly useful in chemical thermodynamics to predict reaction feasibility and understand energy flow.

Understanding total entropy change is crucial for predicting the direction and feasibility of chemical reactions and physical processes. It underpins the concept of spontaneity and equilibrium, guiding the design of efficient chemical processes and understanding natural phenomena from biological systems to cosmological evolution.

Forgetting to convert units (e.g., kJ to J). Incorrectly calculating \Delta S_{surroundings} or using the wrong sign.