What Is r in the Free Energy Equation?

This calculator works out the reaction quotient, Q, for a chemical reaction from reactant and product concentrations and their balanced-equation coefficients. The symbol r is not a standard variable in the usual Gibbs free-energy equation; it may be a misreading or informal label for R, the gas constant, or confusion with Q. Entering concentrations here gives Q, which is the quantity used to find the reaction free-energy change when the other required data are known.

Formula

For a reaction written as nr R → np P, the simple concentration form of the reaction quotient is Q = [P]^np / [R]^nr. Here [P] and [R] are the product and reactant concentrations in mol/L, and np and nr are their stoichiometric coefficients from the balanced equation. The quotient is dimensionless when concentrations are expressed relative to the standard concentration of 1 mol/L; this calculator uses their numerical values in mol/L.

For a general reaction aA + bB ⇌ cC + dD, Q = ([C]^c[D]^d)/([A]^a[B]^b). The calculator displays a two-species version to make the role of each input clear. For reactions with more species, include every product term in the numerator and every reactant term in the denominator, each raised to its balanced coefficient. Pure solids and pure liquids are omitted.

Worked Example

Using the calculator’s default values, the illustrative reaction is R ⇌ P, with both coefficients equal to 1 and both concentrations equal to 0.1 mol/L.

• Numerator: [P]^np = (0.1)¹ = 0.1.
• Denominator: [R]^nr = (0.1)¹ = 0.1.
• Reaction quotient: Q = 0.1 / 0.1 = 1.0000.

The result is dimensionless. It is an example only: the reaction, coefficients, and actual concentrations for a real system must come from its balanced equation and data.

Assumptions

• The two-species example represents a balanced reaction with one product and one reactant; real reactions may have more terms.
• Concentrations approximate thermodynamic activities. This is most reasonable for sufficiently dilute solutions; concentrated or strongly nonideal mixtures require activity coefficients.
• Inputs are positive concentrations. A zero concentration makes the quotient zero or undefined depending on its position and may not represent a usable equilibrium calculation.
• This calculator computes Q only. It does not calculate ΔG without the reaction’s standard free-energy change and temperature.

What r Means in the Free-Energy Equation

The familiar equation for a reaction’s Gibbs free-energy change is ΔG = ΔG° + RT ln Q. In this equation, R is the universal gas constant, T is absolute temperature in kelvins, and Q is the reaction quotient. The conventional symbol is an uppercase R, not a lowercase r. R has the exact value 8.31446261815324 J/(mol·K) in SI units.

There is no single standard meaning for a lowercase r in this equation. If a source prints “r,” it may be using a nonstandard notation, a lowercase rendering of R, or a transcription error. In other contexts r can denote a rate, radius, or another quantity, but those meanings do not make it the gas constant in this formula. Do not substitute an arbitrary r value: check the source’s definitions and equation.

The equation is sometimes written with ΔG = ΔG° + RT ln Q, and sometimes as ΔG = ΔG° + RT ln([products]/[reactants]) for a simple reaction. That second form is shorthand: stoichiometric powers are required, and a rigorous treatment uses activities. R is a constant; Q changes with the current mixture.

How Q Connects to Free Energy

Q describes the composition at the moment being considered. The equilibrium constant K is the value of the reaction quotient at equilibrium, for a specified temperature and standard-state convention. Substituting K at equilibrium gives ΔG = 0, so ΔG° = −RT ln K. Away from equilibrium, compare Q with K:

• If Q < K, then ΔG is negative in the forward direction under the stated conditions; the forward reaction is thermodynamically favored toward equilibrium.
• If Q > K, then ΔG is positive in the forward direction; the reverse direction is favored toward equilibrium.
• If Q = K, the system is at equilibrium and ΔG = 0 for the reaction as written.

“Favored” does not say how quickly a reaction occurs. Reaction rates depend on kinetics, including activation barriers and mechanism. Nor does the sign alone guarantee that a reaction will proceed detectably on a particular timescale.

Common Notation and Calculation Errors

• Confusing R and r: use R for the gas constant in RT. A lower-case r has no universal role there unless a particular source defines it.
• Using coefficients incorrectly: coefficients become powers in Q. For 2A ⇌ B, Q = [B]/[A]², not [B]/[A].
• Using the wrong reaction direction: reversing the equation inverts Q and K and changes the sign of ΔG for the written direction.
• Mixing units: use temperature in kelvins when multiplying by R in J/(mol·K). The concentration quotient must be formed consistently with the standard-state convention.
• Including pure solids or liquids: their activities are treated as 1 and they do not appear as concentration factors in the usual equilibrium expression.
• Calling Q the equilibrium constant: Q uses current composition; K is its equilibrium value at a stated temperature.

For a full free-energy calculation, also make sure ΔG° and RT use compatible energy units. If ΔG° is in kJ/mol, convert RT from J/mol to kJ/mol by dividing by 1000 before adding. The logarithm is the natural logarithm, ln, not log base 10.

Frequently Asked Questions

Is r the gas constant in ΔG = ΔG° + RT ln Q?
No. The standard symbol is uppercase R, the gas constant. A lowercase r is nonstandard in this equation unless a specific source defines it.

What does R equal in the free-energy equation?
In SI units, R = 8.31446261815324 J/(mol·K). Pair it with absolute temperature in kelvins so RT has units of energy per mole.

What is Q in the free-energy equation?
Q is the reaction quotient: a ratio of product activities to reactant activities, each raised to its stoichiometric coefficient. It represents the mixture’s current composition.

Is Q the same as K?
No. Q is evaluated for the current mixture; K is the equilibrium value of that quotient at a particular temperature. They are equal when the system is at equilibrium.

Why are there powers in the reaction quotient?
The powers come from stoichiometric coefficients in the balanced reaction. For example, 2A ⇌ B gives Q = [B]/[A]² in the concentration approximation.

Does negative ΔG mean a reaction is fast?
No. Negative ΔG indicates thermodynamic favorability in the written direction under the stated conditions, not reaction speed. Speed is a kinetic question.

Related

• Percent Yield Calculator: Actual Yield ÷ Theoretical Yield × 100
• Mole Calculator From a Chemical Equation: Moles of Product From the Mole Ratio

in cp | Product concentration | mol/L | 0.1 | 0
in cr | Reactant concentration | mol/L | 0.1 | 0
in np | Product stoichiometric coefficient | - | 1 | 0
in nr | Reactant stoichiometric coefficient | - | 1 | 0
out q | Reaction quotient Q | - | (cp ^ np) / (cr ^ nr) | 4