a. Without using a calculator, estimate the pH of each of the ...

Question

a. Without using a calculator, estimate the pH of each of the following solutions:

1. [HO⁻] = 3.2 × 10⁻⁵

2. [H₃O⁺] = 8.3 × 10⁻¹

3. [H₃O⁺] = 1.7 × 10⁻³

b. Determine the exact pH, using a calculator.

Accepted Answer

Welcome back, everyone estimate the P H of each solution given below based on hydro num or hydroxide concentration without using a calculator. Number one, the concentration of hydroxide is 2.4 multiplied by 10 to the power of negative ninth molar. Number two, the concentration of hydro is equal to 3.5 multiplied by 10 to the negative second molar. Number three, the concentration of hydro num is equal to 5.7 multiplied by sense. The negative four smaller determine the exact P H by using a calculator to begin this problem. We simply want to recall that the P H is equal to the negative logarithm of base 10 of the concentration of hydro num ions. In addition to that, if we're given the concentration of hydroxide, we can simply say that P H is equal to P K W minus P O H. And since P O H is found similarly as P H or basically the negative logarithm of based sun of the concentration of hydroxide, we can combine the two final equations into one and say that we can also find P H by taking P K W and subtracting the negative log of the concentration of hydroxide, which means that we are adding the logarithm of base 10 of the concentration of hydroxide. So primarily for this problem, we are going to use two equations. One which allows us to directly find the P H by taking the negative log of the concentration of hydro. And the second equation which will allow us to find the P H in terms of the concentration of hydroxide. We have to recall that P K W is 14 at room temperature. Well done. So let's go ahead and try to estimate the P H for each solution. Starting with the first one, we're given the concentration of hydroxide. So just as equation number two suggest P H would be equal to P K double E which in this case would be 14 assuming that the solution that we're looking at is at room temperature. And then we're going to add logarithm of base of the concentration of hydroxide, which is 2.4 multiplied by 10 of the power of negative ninth. We're not going to include any units because we're taking a LA A number and that number must be unit less. Let's recall one of the important properties of logarithms. We have to remember that log of a product of two numbers A N B can be split into log of a plus log off B. So that's the multiplication property that we're going to use to estimate. And in addition to that, let's recall the power rule. Whenever we say log of a race to the power of X, we can essentially bring down the exponent and say that this is equal to X multiplied by log of A. And finally, let's also include another property. If we get, if we have log of 10 raised to the power of X, we can just use the previous property. And in addition to that log of 10 is just one. So overall log of 10 to the power of X would be equal to X because as long as our argument is equal to our base log of 10 will be one because in this case, our base of this log is 10. So now going back to our equation, if we want to estimate that we can say that P H would be equal to 14 plus. Now apply one of our rules, specifically the product rule, we will have log of 2.4 plus log off the power of negative ninth. Now simplifying and specifically, we know that log of 10 to the power of negative ninth will be negative nine. So we have minus nine plus log of 2.4. Now, what we want to do here is just combine the like terms so 14 minus nine that gives us five and that we are adding a value of log of 2.4. And of course, we have to recall that if we're taking a logarithm of a specific argument C as long as C is between a one and 10, then the logarithm will be between zero and the one, we can easily check that by adding our boundary values into our argument and evaluating the results. So log of one will be zero log of 10 will be one. So we can conclude that in order to estimate in this case, we can say that log of 2.4 will be between zero and one, right, meaning if we add five to log of 2.4, it should be between five and six, right? Because the maximum value of a log of 2.4 cannot exceed one. So we can say that for number one, the estimated P H will be between five and six. And for the purposes of this problem, we can actually introduce a table. So it will be easier for us to track our thought process. Meaning for the first part, the estimate that P H is between five and six. Now, we can check that by calculating the exact value, we can essentially use our equation, we can add five and log of 2.4. And according to what we get using a calculator, that would be 5.38. So our estimated peach seems reasonable. And now we can move on to the second one. For the second part, we are going to use the concentration of hydro num. So we are simply taking negative log of base of 3.5 multiplied by since the negative second. So now let's go ahead and simplify using our rules. So first of all, we have a negative sign. So here we have to be a bit more careful, we are going to factor out the negative sign and in we're going to apply the product rule. So that will be a log of 3. plus log of 10 to the power of negative seconds, we still have our negative sign in parentheses. We can simplify the result to log of 3.5. And according to our power rule, for the second term, we will have minus two because we're bringing down the exponent and log of 10 becomes one. Now, if we distribute the negative sign, we will get two minus log of 3.5. And we know that log of 3.5 must be between zero and one. So that means our P H the estimated P H value must be between one and two. We can evaluate the exact value taking a difference between two and log of 3.5, which gives us 1.46. So our estimate looks correct and we are finally ready to move on to the third part. We're given the concentration of hydro num. So we're going to take the negative log of that concentration. In this case, that would be 5.7 multiplied by cent the negative fourth, applying the rules of logarithms and factoring out the negative sign, we will get negative logarithm of 5.7 plus logarithm of 10 to the negative fourth. Now we have a negative log 5. applying the power rule. We get negative four and this distributing the negative sign, we can say that we end up with four minus log 5.7. Once again, we're subtracting a valley which is between zero and one. So that means four minus log 5.7 should be between three and four. So that'll be our estimate between three and four and evaluating the exact value. We end up with 3.24. So once again, our estimate looks quite good. Let's go ahead and label our final answers for part 12 and three. We have estimated our pe values and we have also calculated the exact P H values. Thank you for watching and I hope to see you in the next video.

a. Without using a calculator, estimate the pH of each of the following solutions:
1. [HO⁻] = 3.2 × 10⁻⁵
2. [H₃O⁺] = 8.3 × 10⁻¹
3. [H₃O⁺] = 1.7 × 10⁻³
b. Determine the exact pH, using a calculator.

Key Concepts

pH Scale

Hydronium Ion Concentration

Logarithmic Relationships

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