I don't know how to solve for x in this equation....
1.75 x 10^-5 = (x^2)/(0.10-x)
At first, you should transfer the denominator of the fraction on the right side of the equation to the left side of the equation. Then you get:
(0.10-x) x 1.75 x 10^-5 = x^2
=> (1,75 x 10^-6)-(1.75 x 10^-5 x) = x^2
=> x^2-(1.75x10^-5 x)-1.75 x 10^-6 = 0
This is a polynomial equation of degree 2 since it is of the form ax^2+bx+c=0.
This type of equations has two solutions, namely:
x1 = (-b+sqrt(b^2 - 4ac))/(2a)
x2 = (-b-sqrt(b^2 - 4ac))/(2a)
I went ahead and did the calculations for you. I find that x is 0.001331654593 or x is -0.001331654593. I read your previous post and since you are looking for a concentration of ions, only the positive value matters. So you have 0,133 x 10^-2 mol/l H+ ions. If you want to know the pH of the mixture, you need to take the negative of the 10-based logaritme of this number. This gives you:
-log(0.001331654593) = pH = 6.621333055
so the pH of this solution is about 6.62, which means it is slightly acidic.
I plugged in the answer that you found for x...thank you btw...it comes out to be 1.79 x 10^-5....not 1.75 x 10^-5...is there another way to solve for x?
The answer I gave you is in fact correct. I just had a course in numerical mathematics, and because you are working with very small numbers (in the order of 10^-5), there can be relatively large errors when you plug in the solution for x in the equation. Note however that the solution that was found is probably accurate enough for the results you require.
Let me help somewhat here.
1. Although all those digits are correct, you need not include all of them for the Ka of 1.75 x 10^-5 (generally) is not known any better than three significant figures. Therefore, use 1.33 x 10^-3 as the answer.
2. Is there another way? yes. An easier way to solve this, if you don't want to go through the quadratic equation and the quadratic formula, is to make an assumption that 0.1 - x = about 0.1 (that is, that x is small enough that subtracting it from 0.1 won't make much difference). If you make that assumption, then
0.1-x = 0.1 and we have the equation
x^2/0.1 = 1.75 x 10^-5
x = sqrt (1.75 x 10^-6)
x = 1.32 x 10^-3 = (H^+).
So this avoids the quadratic, the equation is solved in just three steps, and the answers is essentially the same. You should always check the assumption to make sure it is valid. 0.1 - 0.00132 is VERY CLOSE to 0.1 so the assumption is valid). GENERALLY, if the answer obtained with the asumption is within 10% of the value we started with (in this case 0.1), then we are ok BECAUSE THE Ka values often are not known to a more accurate value.
3. The value of the pH needs to be recalculated for the pH is not 6.62.
pH = -log(H^+) = -1.32 x 10^-3 = 2.8794 which rounds to 2.88.
I hope this helps.
this really helped me thank you
To solve for x in the equation 1.75 x 10^-5 = (x^2)/(0.10-x), you can follow these steps:
Step 1: Simplify the equation by multiplying both sides by (0.10-x) to eliminate the fraction:
1.75 x 10^-5 * (0.10-x) = (x^2)
Step 2: Distribute the multiplication on the left side of the equation:
1.75 x 10^-5 * 0.10 - 1.75 x 10^-5 * x = (x^2)
Step 3: Simplify further:
0.175 x 10^-5 - 1.75 x 10^-5 * x = x^2
Step 4: Expand the terms on the right side:
0.175 x 10^-5 - 1.75 x 10^-5 * x - x^2 = 0
Step 5: Rearrange the equation to have all the terms on one side:
x^2 + 1.75 x 10^-5 * x - 0.175 x 10^-5 = 0
Step 6: This is now a quadratic equation. You can solve it by factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula. The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In the equation x^2 + 1.75 x 10^-5 * x - 0.175 x 10^-5 = 0, you can identify a = 1, b = 1.75 x 10^-5, and c = -0.175 x 10^-5. Plug these values into the quadratic formula:
x = (-(1.75 x 10^-5) ± √((1.75 x 10^-5)^2 - 4(1)(-0.175 x 10^-5))) / (2(1))
Step 7: Simplify the equation within the square root:
x = (-(1.75 x 10^-5) ± √(3.0625 x 10^-10 + 0.7 x 10^-10)) / 2
Step 8: Combine like terms within the square root:
x = (-(1.75 x 10^-5) ± √(3.7625 x 10^-10)) / 2
Step 9: Simplify the square root:
x = (-(1.75 x 10^-5) ± 1.94 x 10^-5) / 2
Step 10: Simplify further by dividing each term by 2:
x = -0.875 x 10^-5 ± 0.97 x 10^-5
So the solutions for x are x = -0.875 x 10^-5 + 0.97 x 10^-5 and x = -0.875 x 10^-5 - 0.97 x 10^-5.