If a reaction has a small equilibrium constant, the equilibrium position favors the reactants. In such reactions, a negligible change in concentration may occur if the initial concentrations of reactants are high and the Kc value is small. In such circ*mstances, the equilibrium concentration is approximately equal to its initial concentration. This estimation can be used to simplify the equilibrium calculations by assuming that some equilibrium concentrations are equal to the initial concentrations. However, to make this assumption, the change in the concentration of a weak acid or base, i.e., x, must be less than 5% of its initial concentration. If x is more than 5%, then the quadratic formula needs to be used to solve the equilibrium equation.
Calculation of Equilibrium Concentrations Using an Algebra-Simplifying Assumption
What are the concentrations at equilibrium of a 0.15 M solution of HCN?
Using x to represent the concentration of each product at equilibrium gives this ICE table.
| HCN (aq) | H+ (aq) | CN− (aq) |
| Initial Concentration (M) | 0.15 | 0 | 0 |
| Change (M) | −x | +x | +x |
| Equilibrium Concentration (M) | 0.15 − x | x | x |
Substitute the equilibrium concentration terms into the Kc expression
rearrange to the quadratic form and solve for x
Thus, [H+] = [CN–] = x = 8.6 × 10–6 M and [HCN] = 0.15 – x = 0.15 M.
Note in this case that the change in concentration is significantly less than the initial concentration (a consequence of the small K), and so the initial concentration experiences a negligible change:
This approximation allows for a more expedient mathematical approach to the calculation that avoids the need to solve for the roots of a quadratic equation:
The value of x calculated is, indeed, much less than the initial concentration
and so the approximation was justified. If this simplified approach were to yield a value for x that did not justify the approximation, the calculation would need to be repeated without making the approximation.
This text has been adapted from Openstax, Chemistry 2e, Section 13.4 Equilibrium Calculations.
FAQs
The small x assumption is an approximation that can be used under certain conditions to simplify solving the equilibrium expression and to avoid using the quadratic formula.
When can you ignore x in ice tables? ›
To determine whether x is negligible, compare the magnitude of the last decimal place of the concentration of the acid to the magnitude of the equilibrium constant. If the difference in magnitude is greater than 100, the x may be neglected.
In which one of the following conditions is the small x assumption valid? ›
You can use the "x-is-small" approximation when the equilibrium constant has a value of less than 10^-3. With this condition, you can assume that the concentration of the reactants stays the same throughout the reaction to equilibrium because a very small amount is being consumed to make products.
What is the small KC assumption? ›
A small value of Kc indicates that the reaction at equilibrium favors the formation of reactants rather than products. Because it is small, the extent to which the reactants are consumed is considered negligible.
What is the formula for small approximation? ›
For a small angle 𝑥 measured in radians, we can approximate the outputs of the trigonometric function as follows: s i n 𝑥 ≈ 𝑥 , c o s 𝑥 ≈ 1 − 𝑥 2 , t a n 𝑥 ≈ 𝑥 .
How big does the angle x need to be for the small angle approximation to become inaccurate? ›
A small angle is usually less than or equal to 15 degrees or 0.26 radians. To use small-angle approximation the angle has to be in this range. As the angle gets bigger than 15 degrees (0.26 radians), the approximations become less accurate.
What is the 5% rule in chemistry? ›
The 5% approximation rule is a guideline used in chemistry to simplify calculations involving weak acids and bases. It states that if the ionization of a weak acid or base is less than 5%, then the concentration of the un-ionized species can be approximated as equal to the initial concentration.
How to check if x is small? ›
The "x-is-small" approximation is based on the idea that if the system can be arranged so it starts “close” to equilibrium, then if the change (x) is small compared to any initial concentrations, it can be neglected. Small is usually defined as resulting in an error of less than 5%.
When can you use small change assumption? ›
This estimation can be used to simplify the equilibrium calculations by assuming that some equilibrium concentrations are equal to the initial concentrations. However, to make this assumption, the change in the concentration of a weak acid or base, i.e., x, must be less than 5% of its initial concentration.
Why can we ignore a small X in the first case but not in the second? ›
Explain why we can ignore a small x in the first case but not the second. When adding or subtracting a small value of x, we can often ignore x because of significant figures. In addition and subtraction, the number of significant figures depends on the number with the fewest decimal places.
Let "x" represent the change in concentration of the hydrogen gas. Since we start with HI alone, the reaction must proceed to the right resulting in an increase in the amount of hydrogen gas. Example: 0.600 moles of NO and 0.750 moles of O2 are placed in an empty 2.00 L flask.
What is the rule of approximation? ›
The key is to look at the number after the required number of decimal places. e.g. write 7.7186 to 3 decimal places(d.p.) The 6 is what is called the 'decider'. If this number is '5' or more, then the 3rd decimal is 'rounded up'. Less than '5' and the decimal stays the same.
What is the assumption rule in chemistry? ›
We divide x by the initial concentration and multiply by 100. If the answer is less than 5%, the assumption is valid. If it is 5% or greater, the assumption is not valid and the quadratic formula must be used to calculate x.
How to determine if x is small? ›
The "x-is-small" approximation is based on the idea that if the system can be arranged so it starts “close” to equilibrium, then if the change (x) is small compared to any initial concentrations, it can be neglected. Small is usually defined as resulting in an error of less than 5%.
How do you know if over or under approximation? ›
Students should see that in general the linear approximation will overestimate if the curve is concave down and it will underestimate when the curve is concave up.
How do you find the approximation? ›
We can use the linear approximation of a function f(x) to find the values of f(x) at the nearest values of a fixed number x = a. The linear approximation is denoted by L(x) and is found using the formula L(x) = f(a) + f '(a) (x - a), where f '(a) is the derivative of f(x) at a x = a.
What is the small number approximation? ›
The approximation is sometimes called "the law of small numbers" because it approximates the distribution of the number of successes when the chance of success is small: you only expect a small number of successes. As an example, here is the binomial (1000,2/1000) ( 1000 , 2 / 1000 ) distribution.
