If we assume that x is very small compared to 0.00250, the first iteration gives x=(0.00250×4.5×10−4)1/2=0.00106. Even though the assumption is not valid, it seems likely that [HNO2] is closer to (0.00250−0.00106)M than to 0.00250 M. Substitute the new value into the original equation to obtain the second iteration,
x20.00250−0.00106=4.5×10−4
and then x=8.04×10−4. Now, we can restrict x to the following range:
8.04×10−4<x<1.06×10−3
Use the new value of x to write that [HNO2]=(0.00250−0.000804)M and solve the equation once again for x to obtain the third iteration,
x20.00250−0.000804=4.5×10−4
which yields x=8.74×10−4. Now, we can further restrict x to the range
8.04×10−4<x<8.74×10−4
Continue this procedure until x attains a constant value. The values of x for the next three iterations are shown in the following table.
Iteration | Value of x |
4 | 8.54×10−4 |
5 | 8.60×10−4 |
6 | 8.59×10−4 |
When the value of x (rounded to two significant figures) becomes constant, such as in iterations 5 and 6, the process ends. In this case, we can say that, to two significant figures, [H3O+]=8.6×10−4M.
Part A
Now you will solve the same problem as above, but using the quadratic formula instead of iterations, to show that the same value of x is obtained either way.
Using the quadratic equation to calculate [H3O+] in 0.00250 M HNO2, what are the values of a, b, c and x , where a, b, and c are the coefficients in the quadratic equation ax2+bx+c=0, and x is [H3O+]? Recall that Ka=4.5×10−4 .
Express a, b, c, and x numerically separated by commas.
Part B
Using the method of successive approximations, what is [H3O+] in 0.500 M HClO2 (Ka=1.1×10−2) and how many iterations (n) are needed to validate that a constant value had been obtained to two significant figures?
Express [H3O+] using two significant figures, followed by a comma and the number of iterations as an integer.
Part C
Using the method of successive approximations, what is [H3O+] in 0.00500 M CH2ClCOOH(aq) (Ka=1.4×10−3) and how many iterations (n) are needed to validate that a constant value had been obtained to two significant figures?
Express [H3O+] using two significant figures, followed by a comma and the number of iterations as an integer.
If we assume that x is very small compared to 0.00250, the first iteration gives x=(0.00250×4.5×10−4)1/2=0.00106....
Using the method of successive approximations, what is [H3O+][H3O+] in 0.500 M M HClO2HClO2 (Ka=1.1×10−2Ka=1.1×10−2) and how many iterations (nnn) are needed to validate that a constant value had been obtained to two significant figures? Express [H3O+][H3O+] using two significant figures, followed by a comma and the number of iterations as an integer.
Using the method of successive approximations, what is [H3O+] in 0.00500 M CH2ClCOOH(aq) (Ka=1.4×10−3) and how many iterations (n) are needed to validate that a constant value had been obtained to two significant figures? Express [H3O+] using two significant figures, followed by a comma and the number of iterations as an integer.
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