X is small approximation

The approximation is therefore vahd. In Example 14.12 and 14.13, we treat two nearly identical problems—the only difference is the initial concentration of the reactant. In Example 14.12, the initial concentration of the reactant is relatively large, the equilibrium constant is small, and the x is small approximation works well. In Example 14.13, however, the initial concentration of the reactant is much smaller, and even though the equihb-rium constant is the same, the x is small approximation does not work (because the initial concentration is also small). In cases such as this, we have a couple of options to solve the problan. We can either solve the equation exactly (using the quadratic formula, for example), or we can use the method of successive approximations, which is introduced in Example 14.13. In this method, we essentially solve for x as if it were small, and then substitute the value obtained back into the equation (where x was initially neglected) to solve for x again. This can be repeated until the calculated value of x stops changing with each iteration, an indication that we have arrived at an acceptable value for x. 

Note that the x is small approximation does not imply that x is zero. If that were the case, the reactant and product concentrations would not change from their initial values. The X is small approximation just means that when x is added or subtracted to another number, it does not change that number by very much. Eor example, we can calculate the value of the difference 1.0 - x when x = 3.0 X 10 ... 

As is often the case with equilibrium problems, we arrive at a quadratic equation in Jc, which we can solve using the quadratic formula (see Appendix IC). However, in many cases we can apply the x is small approximation (first discussed in Section 14.8). In Examples 15.5 and 15.6, we examine the general procedure for solving weak acid equilibrium problems. In both of these examples, the x is small approximation works well. In Example 15.7, we solve a problem in which the x is small approximation does not work. In such cases, we can solve the quadratic equation explicitly, or apply the method of successive approximations (also discussed in Section 14.8). Einally, in Example 15.8, we work a problem in which we find the equilibrium constant of a weak acid from its pH. 

In many cases, you can make the approximation that x is small (as discussed in Section 14.8). Substitute the value of the acid ionization constant (from Table 15.5) into the Ka expression and solve for x. Confirm that the x is small approximation is valid by calculating the ratio of x to the number it was subtracted from in the approximation. The ratio should be less than 0.05 (or 5%). 

Check your answer by substituting the computed equilibrium values into the acid ionization expression. The calculated value of A a should match the given value of Kg. Note that rounding errors and the x is small approximation could result in a difference in the least significant digit when comparing values of Kg. ... 

EXAMPLE 15.7 Finding the pH of a Weak Acid Solution in Cases Where the x is small Approximation Does Not Work Find the pH of a 0.100 M HCIO2 solution. SOLUTION ... 

Check to see if the x is small approximation is valid by calculating the ratio of x to the number it was subtracted from in the approximation. The ratio should be less than 0.05 (or 5%). 

The initial concentration and K s, of several weak acid (HA) solutions are listed here. For which of these is the x is small approximation least likely to work in fmding the pH of the solution ... 

Use the definition of percent ionization to calculate it. (Since the percent ionization is less than 5%, the x is small approximation is valid.)... 

Snbstitute the expressions for the eqnilibrinm concentrations (from the table just shown) into the expression for In this case, you cannot make the x is small approximation because the equilibrium constant (0.012) is not small relative to the initial concentration (0.0100). 

Notice from the results of Example 15.19 that the concentration of X for a weak diprotic acid H2X is equal to A aj. This general result applies to all diprotic acids in which the X is small approximation is valid. Notice also that the concentration of H30 produced by the second ionization step of a diprotic acid is very small compared to the concentration produced by the first step, as shown in Figure 15.13 t. 

When calculating [H30 for weak acid solutions, we can often use the x is small approximation. Explain the nature of this approximation and why it is valid. 

Calculate the [H30 ] and pH of each H2SO4 solution. At approximately what concentration does the x is small approximation break down ... 

If we make the same x is small approximation that we make for weak acid or weak base equilibrium problems, we can consider the equilibrium concentrations of HA and A to be essentially identical to the initial concentrations of HA and A (see step 4 of Example 16.1). Therefore, to determine [H30 ] for any buffer solution, we multiply by the ratio of the... 

Substitute the expressions for the equilibrium concentrations into the expression for the acid ionization constant. Make the x is small approximation and solve for x. 

We then substitute the expressions for the equilibrium concentrations into the expression for the acid ionization constant. As long as is sufficiently small relative to the initial concentrations, we can make the x is small approximation and solve for x, which is equal to [H30 ]. 

Part II Equilibrium Alternative (using the Henderson-Hasselbalch equation). As long as the x is small approximation is valid, you can substitute the quantities of acid and conjugate base after the addition (from part I) into the Henderson-Hasselbalch equation and calculate the new pH. 

Notice that, after the addition, the solution contains significant amounts of both an acid (HCHO2) and its conjugate base (CH02 ) —the solution is now a buffer. To calculate the pH of a buffer (when the x is small approximation applies as it does here), we can use the Henderson-Hasselbalch equation and pK for HCHO2 (which is 3.74) ... 

Substitute the equihbrium expressions for [Ca ] and [F ] from the previous step into the expression for Since is small, you can make the approximation that 2S is much less than 0.100 and will therefore be insignificant when added to 0.100 (this is similar to the X is small approximation in equilibrium problems). A,p = [Ca +][F f = 5(0.100 + W (S is small) = 5(0.100)2... 

Henderson-Hasselbalch equation An equation used to easily calculate the pH of a buffer solution from the initial concentrations of the buffer components, assuming that the x is small" approximation [base]... 

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