Algorithm title: POJ 2006 Litmus Test (simple math & ionization constants of weak acid)

Litmus Test

http://poj.org/problem?id=2006

Time limit:1000ms

Memory limit:30000k

Description

The PH scale measures the concentration of protons (H⁺) in a solution and, therefore, its acidity or alkalinity. The PH value of a solution is a number between 0 and 14; It is less than 7 if the solution is acidic, greater than 7 if the solution are basic, and 7 if it is neutral.

The formula for calculating PH

PH =-log [H ⁺]

where [H⁺] is the concentration of protons measured in moles per litre.

To calculate the PH value of a acid, one has to determine the concentration of protons in the solution. When a acid is dissolved in water, a equilibrium is reached and are governed by the equation

K _a = [H ⁺] [acid ions]/[acid]

Where K_a is the acidity constant (known for all acid), [acid ions] is the concentration of the acid ions that ha ve dissolved, and [acid] is the concentration of the undissolved acid. Before the acid is added, both [H⁺] and [acid ions] are assumed to be 0.
For example, the acidity constant of methanoic acid is 1.6 *⁴. Dissolving one mole of acid molecules results in one mole of H⁺ and one mole of acid ions. If the initial concentration of the methanoic acid is 0.1 moles/l and x moles of acid are dissolved (per liter), then the Final concentration at equilibrium would is 0.1-x moles/l for the acid and x moles/l for H⁺ and the acid ions.

Input

The input consists of a number of test cases. Each test case contains 4 numbers on a line:two positive floating-point numbers specifying the acidity constant K_{a and the original concentration of the acid (in Moles/liter) added to the water, as OK as two positive and integers m and n indicating that each mole of acid molecules are dissolved into m moles of H⁺ ions and n moles of acid ions. The floating-point numbers are specified in scientific notation as shown. The input is terminated with a line containing four zeros.}

Output

For each test case, print on a line the PH value of the solution, rounded to 3 decimal places.

Sample Input

1.6e-04 1.0e-01 1 1 1.6e-04 1.0e-01 4 1 1.5e-05 5.0e-02 1 2 0 0-0 0

Sample Output

2.407
2.101
3.216

If the acid is ionized by the X mol, then there is a ka= (MX*NX)/(C-x), and an x is multiplied by M that is [h+] ([matter] indicates the concentration of the substance)

Complete code:

/*110ms,192kb*/
  
#include <cstdio>
#include <cmath>
  
int main (void)
{
    double ka, acid ;
    int m, n;
    while (scanf ("%lf%lf%d%d", &ka, &acid, &m, &n), m)
        printf ("%.3f\n",-log10 (-ka + sqrt (ka * ka + (M &l t;< 2) * n * acid 

* ka))/(n << 1));
    return 0;
}

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