Kp Calculator: Equilibrium Constant for Gas Reactions

What is Kp?

Kp is the equilibrium constant for a chemical reaction that happens between gases, written in terms of partial pressures. A Kp calculator takes the partial pressures and coefficients from a balanced equation and returns this number. Kp above 1 means the reaction favors products at equilibrium. Kp below 1 means it favors reactants.

Chemists use Kp instead of the concentration-based constant Kc whenever pressure is the natural way to measure how much gas is present, such as in industrial reactors or the atmosphere.

Kp formula

For a general reaction

$aA + bB \rightleftharpoons cC + dD$

the equilibrium constant is

$K_p = \frac{(P_C)^c \times (P_D)^d}{(P_A)^a \times (P_B)^b}$

Each partial pressure ($P$) is raised to the power of its stoichiometric coefficient (the number in front of that substance in the balanced equation). Products go on top, reactants go on the bottom.

A few rules apply:

• Pressures must be positive numbers, usually measured in atmospheres (atm). Any consistent pressure unit works, as long as the same unit is used throughout one calculation.
• Pure solids and liquids are left out of the expression, because their activity is fixed at 1. For CaCO₃(s) ⇌ CaO(s) + CO₂(g), only the CO₂ pressure appears, so $K_p = P_{CO_2}$.
• Kp is only valid at one temperature. Changing the temperature changes Kp; changing the total pressure does not.

Kp relates to Kc, the concentration-based equilibrium constant, through

$K_p = K_c \times (RT)^{\Delta n}$

where $R = 0.0821$ L·atm/(mol·K), $T$ is the temperature in kelvin, and $\Delta n$ is the moles of gaseous products minus the moles of gaseous reactants. When $\Delta n = 0$, Kp and Kc are numerically equal.

How to calculate Kp with this tool

1. List the reactants. For each one, enter its chemical formula (optional, for reference), its stoichiometric coefficient from the balanced equation, and its partial pressure at equilibrium. Add more rows with "Add Reactant" if the equation has more than one reactant.
2. List the products the same way, using "Add Product" for extra rows.
3. Read the result. The calculator raises each pressure to its coefficient, multiplies the product terms together, multiplies the reactant terms together, and divides. The Kp value updates as soon as every field has a number — there is no separate calculate button.

Very small results (below 0.001) or very large ones (above 10000) are shown in scientific notation, for example 2.5000e-6. Other results are shown as a plain decimal.

Worked example: ammonia synthesis

The Haber process makes ammonia from nitrogen and hydrogen:

$N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)$

Suppose the equilibrium partial pressures in a reaction vessel are:

• $P_{N_2} = 0.5$ atm (coefficient 1)
• $P_{H_2} = 0.2$ atm (coefficient 3)
• $P_{NH_3} = 0.8$ atm (coefficient 2)

$K_p = \frac{(P_{NH_3})^2}{(P_{N_2})^1 \times (P_{H_2})^3} = \frac{(0.8)^2}{(0.5) \times (0.2)^3} = \frac{0.64}{0.004} = 160$

A Kp of 160 means the equilibrium mixture contains far more ammonia than nitrogen and hydrogen.

A second example: water gas shift reaction

$CO(g) + H_2O(g) \rightleftharpoons CO_2(g) + H_2(g)$

With $P_{CO} = 0.1$ atm, $P_{H_2O} = 0.2$ atm, $P_{CO_2} = 0.4$ atm, and $P_{H_2} = 0.3$ atm, all coefficients equal to 1:

$K_p = \frac{0.4 \times 0.3}{0.1 \times 0.2} = \frac{0.12}{0.02} = 6$

Why Kp matters

Comparing Kp to the reaction quotient Q (calculated from pressures that are not yet at equilibrium) predicts which way a reaction will shift. If Q is less than Kp, the reaction makes more product. If Q is greater than Kp, it makes more reactant.

Industrial processes rely on this. The Haber-Bosch process for ammonia and the contact process for sulfuric acid are both run at temperatures and pressures chosen using Kp data, balancing yield against reaction speed. Kp is also used to model atmospheric equilibria, such as how sulfur dioxide converts to sulfur trioxide in acid rain formation.

Frequently asked questions

What is the difference between Kp and Kc? Kp uses partial pressures; Kc uses molar concentrations. They describe the same gas-phase equilibrium and are related by $K_p = K_c(RT)^{\Delta n}$.

Does changing the total pressure change Kp? No. Kp depends only on temperature. Changing pressure shifts the equilibrium position — the individual partial pressures adjust — but the value of Kp itself stays the same at constant temperature.

Can Kp be negative or zero? No. Kp is a ratio of positive pressures raised to positive powers, so it is always positive. A reaction that barely proceeds has a Kp close to zero but never equal to it.

Why don't solids and liquids appear in the Kp expression? A pure solid or liquid has a fixed composition that does not change during the reaction, so its activity is defined as 1 and it drops out of the ratio mathematically.

What does Kp = 1 mean? The product term and the reactant term in the expression are equal. It does not mean the individual pressures are equal, since coefficients affect the calculation.

When should real (non-ideal) gas behavior be considered? Below about 10 atm, most gases behave close enough to ideal for standard Kp calculations. Above that, fugacity (an effective pressure that corrects for intermolecular forces) is usually used instead of the measured pressure.

References

1. Atkins, P. W., & De Paula, J. (2014). Atkins' Physical Chemistry (10th ed.). Oxford University Press.
2. Chang, R., & Goldsby, K. A. (2015). Chemistry (12th ed.). McGraw-Hill Education.
3. IUPAC (2014). Compendium of Chemical Terminology (the "Gold Book").