# Entropy Calculator

Created by Gabriela Diaz
Last updated: Sep 19, 2022

With our entropy calculator, you can determine the entropy change of chemical reactions and the isothermal entropy change of ideal gases. You'll also be able to study a process' spontaneity through the Gibbs free energy equation.

In the accompanying text of this tool, you'll learn:

• What entropy is;
• How to calculate the entropy change of a reaction;
• The entropy change formula for chemical reactions and for an isothermal process; and
• The Gibbs free energy equation.

## Entropy definition

We frequently come across that entropy is referred to as a measure of the disorder of a system. This definition derives from Boltzmann's views on entropy and the second law of thermodynamics.

Boltzmann's formulation of entropy (around the year 1872) studies it from a microscopic point of view and sees it as a probabilistic phenomenon. According to Boltzmann's model, entropy is related to the number of possible arrangements a system's particles can have.

These possible arrangements or configurations are known as microstates, and the greater the number of microstates, the higher the system's entropy. We can see this in Boltzmann's entropy formula:

$\small S = k_B \cdot \text{ln}(\Omega)$

where:

• $S$ – Entropy of the system, usually in units of $\text{J/K}$;
• $k_B$ – Boltzmann constant, $1.38\times10^{-23} \ \text{J/K}$; and
• $\Omega$ – Total number of possible microstates or multiplicity.

Empirical data shows that the number of disordered states is greater than the ordered ones. Hence, systems will tend to be in a disordered configuration. This is why Boltzmann refers to entropy as a probabilistic phenomenon.

It's worth noting that this microscopic understanding of entropy comes from an earlier macroscopic approach based on heat engine studies conducted by scientists like Carnot and Clausius.

💡 Did you know Carnot created a hypothetical cycle, from which he defined the maximum efficiency that a heat engine can have? You can learn more about this with our Carnot efficiency calculator.

Based on Carnot's work (1824), Clausius was the first to come up with the idea of entropy (during the 1850s) as a way to understand that in the transformations of work to heat and heat to work, heat is not always conserved, as it was previously thought. Clausius' studies resulted in the Clausius inequality for thermodynamic cycles:

$\small \oint \cfrac{\delta Q}{T_{\text{surr}}} \leq 0$

where:

• $\delta Q$ – Small amount of heat supplied; and
• $T_{\text{surr}}$ – Temperature of the surroundings.

In the case of reversible cycles, this integral is equal to zero, $\oint \frac{\delta Q}{T_{\text{surr}}} = 0$. From this case of reversible processes, he derives the concept of entropy and its formula for reversible processes:

$\small {dS}_{\text{rev}} = \left(\cfrac{\delta Q}{T}\right)_{\text{rev}}$

where:

• ${dS}_{\text{rev}}$ – Change of entropy of a reversible process;
• $\delta Q$ – Heat supplied; and
• $T$ – Temperature of the system.

Clausius concluded that the entropy generated $(S_{\text{gen}})$ during an irreversible process is always positive or at least equal to zero, but never negative:

• If $S_{\text{gen}} > 0$ – Irreversible process;
• If $S_{\text{gen}} = 0$ – Reversible process; and
• If $S_{\text{gen}} < 0$ – Impossible process.

## How to calculate the entropy change of a reaction – Entropy change formula

When studying the entropy of a process, we're interested in determining the change of entropy between two states of a system, state 1 and state 2:

$\small \Delta S = S_2 - S_1$

With this entropy calculator, you can determine the entropy change of a chemical reaction. To do so, we just need to take the difference between the products' total entropy and the reactants' total entropy. The formula for entropy change in chemical reactions is:

$\small \Delta S_{\text{reaction}} = \Delta S_{\text{products}} - \Delta S_{\text{reactants}}$

where:

• $\Delta S_{\text{reaction}}$ – Entropy change of the reaction;
• $\Delta S_{\text{products}}$ – Entropy change of the products; and
• $\Delta S_{\text{reactants}}$ – Entropy change of the reactants.

For chemical reactions, we usually express the entropy in its molar units, $\text{J/K} \cdot \text{mol}$.

You can continue learning more about chemical reactions with our activation energy calculator.

## How to find the entropy change of an isothermal process of an ideal gas

You can use this entropy calculator to determine the entropy change of an ideal gas in isothermal processes. For isothermal processes, or processes that happen at constant temperature $(\Delta T=0)$, we can simplify any of the forms of the entropy differential formulas:

\small \begin{aligned} dS &= c_v \cfrac{dT}{T} + n \ R\cfrac{dV}{V} \\ dS &= c_p \cfrac{dT}{T} - n \ R\cfrac{dP}{P} \end{aligned}

Since we're studying isothermal process, $dT = 0$:

\small \begin{aligned} dS &= n \ R\cfrac{dV}{V} \\ dS &= -n \ R\cfrac{dP}{P} \end{aligned}

After integrating between states 1 and 2, we get two equivalent expressions from which we can determine the entropy change for an ideal gas undergoing an isothermal process:

\small \begin{aligned} \Delta S &= S_2 - S_1 = n \ R \ \text{ln}(V_2/V_1) \\ \Delta S &= S_2 - S_1 = -n \ R \ \text{ln}(P_2/P_1) \end{aligned}

where:

• $\Delta S$ – Entropy change of the gas;
• $n$ – Number of moles of the gas;
• $R$ – Ideal gas constant, $8.3145 \text{ J/mol}\cdot \text{K}$.;
• $V_2$ and $V_1$ – Final and initial volume of the system; and
• $P_2$ and $P_1$ – Final and initial pressure of the gas.

Would you like to learn more about ideal gases and the ideal gas law? then you should check our ideal gas law calculator.

## Gibbs free energy equation

With this entropy calculator, you'll also be able to determine the change in Gibbs free energy using the results for entropy change that you get from other sections of the calculator. The Gibbs free energy equation is:

$\small \Delta G = \Delta H - (T \cdot \Delta S)$

where:

• $\Delta G$ – Gibbs free energy change;
• $\Delta H$ – Entropy change;
• $T$ – Temperature in Kelvin; and
• $\Delta S$ – Entropy change.

We use the change in Gibbs free energy to study the spontaneity of a process; this is, whether or not a process requires outside energy to occur. When calculating the change of Gibbs free energy, we'll always encounter any of these three possibilities:

• $\Delta G < 0$ – Spontaneous process;
• $\Delta G = 0$ – System at equilibrium; and
• $\Delta G > 0$ – Non-spontaneous process; additional energy must be put in for the reaction to happen.

## What does entropy represent? – Quality of the energy

Up to this point, we've gone through entropy's definition, formula, and applications, but if you're still unsure what it represents, you're not alone. In contrast to energy, entropy can feel like an abstract concept, not so simple to grasp.

In order to get a more intuitive idea of what entropy represents, it can be helpful to see it as a measure of the quality of the energy. This implies that we can distinguish between "high-quality" and "low-quality" forms of energy.

Which are forms of high-quality energy? Those that are readily available to be used, as is the case of chemical energy stored in a battery, electrical energy, mechanical energy, and some fossil fuels. On the other hand, we consider heat at low temperatures to be a low-quality form of energy, as this can produce little to no work.

As a thought exercise, we can think of a battery-operated lamp. The clumped energy in the batteries is ready to be used at any time to turn on the light bulb. When you turn on the lamp, you'll notice the bulb is heating up, and this heat is lost to the surroundings.

If we apply the first law to this system, we'll see that the energy is conserved. However, when calculating the entropy generation, this will be positive. We can see this as the initial electrical energy from the batteries degrading into heat, which we won't be able to use or reverse into electrical energy.

From the above, you can see that the statement: "the entropy of the universe is constantly increasing" indicates that the universe's energy is gradually degrading from high to low quality.

At some point, the total amount of energy will be the same, but unable to generate work. We can see how those high-quality sources of energy that we still have are extremely valuable, and it's critical to protect them.

Gabriela Diaz
Entropy change for a reaction
Total entropy of products
J
/mol*K
Total entropy of reactants
J
/mol*K
Entropy change for a reaction
J
/mol*K
Gibbs free energy ΔG = ΔH - T*ΔS
Change in enthalpy
J
Temperature
°F
Change in entropy
J
/K
Change in Gibbs free energy
J
Isothermal entropy change of an ideal gas
Base variable
volume
Amount of moles
mol
Initial volume
Final volume
Change in entropy
J
/K
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