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How do you calculate log and -log without a calculator

Discussion in 'DAT Discussions' started by yh4747, Sep 24, 2014.

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  1. yh4747

    yh4747 2+ Year Member

    Aug 19, 2014
    Please help! How would you calculate pH without a calculator. For example, how would you calculate -log(0.1).
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  3. mkhan27

    mkhan27 5+ Year Member

    Mar 18, 2011
    This is best taught with Chads videos. Definitely check his QR section and GC section. I'll try my best in explaining this stuff, but I cannot say that I am the genius here.

    So in regular math, normally log is a standard with a base 10. So log (1) = 0. Anything below 1 would be a negative number and anything above 1 would be a positive number. This is the case for log, NOT -log. Some examples: log(0.1)= -1. Its less than 1. Think of it this way: log (1) is 0 because we are at 1. Now when we move 1 decimal place to the left from 1.0 ---> 0.10, then we have log (0.1)= -1. Also, 0.1 is the same thing as 1x10^-1, so if you want to take the log of that then it would be -1 (just use the exponent, which shows its -1). So how about log (0.01).....So 0.01 can also be written as 1x10^-2. We see that this number is 2 decimals to the left of 1.0, so its will be -2. How about log(10) also written as 1x10^1. So we see that this number is 1 decimal place to the right of 1.0, which means that its going to be 1. How about log(100) also written as 1x10^2, which is 2 decimal places to the right of 1.0, which means that its going to be 2. How about log(1000) also written as 1x10^3, which is 3 decimal places to the right of 1.0, which means that its going to be 3. I hope you are seeing a trend....Let me summ this up below:

    log(0.0001) = -4
    log(0.001) = -3
    log(0.01) = -2
    log(0.1) = -1
    log(1) = 0
    log(10) = 1
    log(100) = 2
    log(1000) =3
    log(10000) =4
    log(1x10^5) =5

    Do you see a trend here....Okay so we did log, now lets do -log, which is something we do in General Chemistry for calculating pH and pOH. This is something you were asking about so it pertains to your question more. The information above was to give you a little understanding of how it works.

    If we are talking about pH in terms of chemistry, then its a completely different thing. So, Chad says p(the sky is blue) = -log(the sky is blue). This is the standard way to calculate pH or pOH. So pH = -log[H+] and pOH = -log[OH-]. So if we had an example of a solution that is 1x10^-12 M and we were asked what the pH would be, then we would do -log(1x10^-12). Here we can use the exponent so the pH would be 12. Think of 1x10^-12 as 0.000000000001, where we are 12 decimal places to the left of 1.0. Since we are taking the -log of the -12, it will be come a +12. This is the way it works in GChem.

    One thing to remember is if the [H+] increases then the pH decreases (becomes more acidic). If the [H+] decreases, then the pH increases (becomes less acidic). This is the same for OH- and pOH. If the [OH-] increases then the pOH decreases (which means that the pH is increasing and the solution is becoming more basic) and if the [OH-] decreases, then the pOH increases (which means that pH is decreasing so the solution is becoming less basic and more acidic). Remember as pH gets bigger the pOH gets smaller and vice versa. If the pH gets smaller the pOH gets bigger. In GChem we always use the pH as our standard for telling if the solution is acidic or basic. So we will use pH and then I will put an example for pOH later.

    So, let me ask you what is the pH of a solution of with a H+ concentration of 1x10^-4. So we take the -log of this. So -log(1x10^-4). Look at the exponent of -4 and that is a complete giveaway that its going to be 4. Remember we are taking the -log so the negative and negative will become a positive. How about: What is the pH of a solution with a H+ concentration of 1x10^-6. So look at the exponent again and that is a giveaway that its going to be a pH of 6 (remember again we are taking the -log. Let's make this a little tricky now....

    What is the pH of a solution that has a H+ concentration of 2x10^-6. So we are not using the 1x10^-6 standard anywhere. Here we have 2x10^-6, which changes things up. First, you already know that 2 > 1. This means that a H+ concentration of 2x10^-6 > 1x10^-6. This means that there is more H+ ions in the 2x10^-6 solution than if it was a 1x10^-6 solution. A solution with more H+ ions will be more acidic and anything that is more acidic will have a lower pH. So 2x10^-6 has to be more acidic than 1x10^-6. We can use our understanding from the example above that a solution of 1x10^-6 H+ concentration has a pH of 6. Since 2x10^-6 is more acidic and has a greater H+ concentration than 1x10^-6, it means that the pH of 2x10^-6 has to be lower than 6 because its more acidic. We also know from Chads videos that 3.16x10^(-n-1) has a pH of n.5. This means that at 3.16 we are at the halfway point (i.e. 5.5, 6.5, 7.5, 8.5, etc). Since 2 < 3.16, this means that 2x10^-6 < 3.16x10^-6, which means that the pH of the 2x10^-6 H+ solution has a pH above 5.5, but below 6.0 (again because its more acidic than 1x10^-6). So the pH would most likely be ~5.8 for a 2x10^-6 H+ solution.

    Lets try a 3.16 example: What is the pH of a solution with a H+ concentration of 3.16x10^-4. We said above that 3.16x10^(-n-1) has a pH of n.5. In this example the n=-4. So, if we took the -log(3.16x10^-4), we can find the pH. One thing to note is that find the pH simply by looking at the exponent. In fact looking at the exponent first will tell you where you have to be. That is the first thing you should do because it can help narrow down answer choices. So, we see that the exponent is around 4, which means that the pH has to be ~4. BUT we also see that the H+ concentration is 3.16, which means that its a halfway point (i.e. 0.5). We said earlier that 3.16 > 1.0, which means that 3.16x10^-4 > 1.0x10^-4, which means that the 3.16x10^-4 has a greater H+ concentration and will have a more acidic solution (i.e. lower pH). We also mentioned above that 3.16 is the halfway point so the halfway point being less than pH 4 would be 3.5. And indeed the pH of a 3.16x10^-4 H+ solution has a pH of 3.5.

    Let me give you a tougher one now...What is the pH of a solution with the H+ concentration of 9x10^-3. So this is more acidic than 1x10^-3. A 1x10^-3 H+ solution would have a pH of exactly 3, but for 9x10^-3 the H+ concentration is greater, which means that its going to be more acidic and will have a pH lower than 3. One thing we see here is that 9x10^-3 is very close to 10x10^-3, which is the same thing as 1x10^-2. This means that the pH will be around 2, BUT it will be above 2.0 because we are at 9x10^-3, which means that its not quite there at 2.0. Most likely the pH of a 9x10^-3 H+ solution would be ~2.1-2.2. What if I asked you what the pOH of this solution would be? You should know that pH + pOH =14. So to calculate we do pOH = 14 - pH -----> pOH = 14-2.1 ===> 11.9. So the pOH is 11.9. To calculate the [OH-] of a solution with a pOH of 11.9 we would see that it would be 1x10^-12 IF the pOH was exactly 12, but in this case its its less than 12, which means that the OH- concentration is around 2x10^-12.

    Lets throw a curveball here.....What is the pH of a 0.2 M NaOH solution? So here we are given a base...a strong base (NaOH). So the concentration we are given is the OH- concentration. So we can use this to get our pOH from it and then use the pH + pOH = 14 calculation to get our pH from there. So pOH = -log(0.2) => this will most likely have a pOH from less than 1. Remember we said that -log(0.1) or -log(1x10^-1) =1. Since 0.2 > 0.1 we will have a lower value. We also stated above that as the concentration increases, then the pH/pOH will be lower. So here the OH- concentration is greater so the pOH will be lower than 1. In fact the pOH= ~0.8. Now we can use pH = 14- pOH -----> 14 - 0.8 ----> 13.2. So the pH=13.2 This is a very basic solution and indeed it should be because NaOH is a strong base.

    One other thing to remember is that for solutions like a 0.1 H2SO4 the pH will be less than 1 because it has 2 H's in H2SO4 (so it will be more acidic than a pH of 1). Its pH will probably be ~0.9. Also for things like Ba(OH)2 or Ca(OH)2 we have 2 OH's. So if we had a 1x10^-3 solution of Ca(OH)2 then the pOH will be less than 3. We can do this by doing 2(1x10^-3). So we multiplied the concentration by 2 because we had 2 moles of OH. This gives us a concentration of 2x10^-3 OH- in the solution. Now taking the -log(2x10^-3) we get a pOH of 2.8. So the pH = 14-2.8 => 11.2, which means that its more basic than if it was a single OH solution (like in NaOH or KOH where we have only 1 OH-).

    I hope this stuff helps. And let me know if you have any questions. Other people will definitely put stuff too and that will also be useful. My recommendation though is to make sure you go through those Chads videos. Even if you watched the video and didn't understand it, then re-watch it again and pay attention to all the details. There must have been something you missed if you are confused. Do his quizzes immediately after the video, but make sure you do them later as well to reinforce material you are learning and have learned. You will do well in Gchem and Ochem with Chads videos, so definitely utilize every bit of it.
    Brightstar17 and nattmelson like this.

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