Mixing Hot & Cold Water

by Patrick Ford
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Hey everybody. So welcome back. Hopefully got a chance to try this one on your own. So we're gonna be mixing two cups of water together. So let's get started, I'm gonna draw this out first. So we have one cup of water that's 0.5 kg at 15 degrees Celsius. So I've got a cup of water like this, You know it's 15°C and it's going to be 0.5 kg. Now we're gonna add this to another cup of water except this water is gonna be boiling, right? So we've got another, we've got boiling water at 100°C. So we've got some amount of water, we don't know how much it is. All we know is that it's about 100°C. So it's really really really close to boiling And they were going to add them together so that the final mixture is exactly 80°C. So when you combine these two things in a larger container, you're gonna end up with this amount of water here, something larger. And this final mixture here is gonna be 80°C. Now this is a pretty classic kalorama tree type question because we're gonna be combining two or more materials in a container and they're both going to reach some equilibrium temperature. Alright, so we're gonna go ahead and stick to our steps here. The first thing we're gonna do is write Q. A. Equals negative QB. Basically. What's going on here is I've got the heat that the 15C water gains is going to be equal to the heat that the 100 degrees Celsius water loses until they finally reach something that's somewhere in between. So that's the first step here is writing Q. A. Is equal to negative Q. B. Now. Really, what's going on here is I want to work my way towards how much boiling water I need to add so that this mixture is 80°C. So if you think about what's happening here is this is cup a and this is cup be the mass of this water here is 0.5 kg. And what I'm looking for here is m be the mass of the boiling water. That's really what I'm looking towards. Alright, so let's go move on with the second step. Now, which is we're going to replace the cues with them Cats. Right? That's our cue equals M. Cat equation. EMC delta T. So this is going to be M. A, times C. For water, right? And we're using C. For water, that's 41 86. Just in case you forget it. And then times delta A adult T for a. Now this is equal to negative MB, times C. For water, times delta T for B. So really all I have to do here is just move on to the last step, which is salt for my target variable. And that's this M. B. In this case. So let's go ahead and start plugging in some numbers. So what I've got here. One thing I can do is I can actually just cross off the specific heats for water for both sides of the equation and I can only do this because it's the same substance I'm dealing with water and water. So it kind of just goes away from the equation. So that's the first thing I can do and then I'm just gonna start plugging in some numbers. Right? So this is gonna be 0.5. What about the change in the temperature for the water? Well that's always a final minus initial. So this is gonna be the T final for both of them, whereas this is gonna be the T. A. initial for uh for a. And this is gonna be TB initial for B. So really this is gonna be my final temperature of 80 minus my initial temperature of 15. And I also don't have to convert it to kelvin's because when you're working with Delta Tes remember it could be Celsius or kelvin. So that's the equation on the right side, What we have is negative MB. And then for Delta T. B, what we have is final minus initial. So this is going to be 80, that's still the final for both of them - the initial of 100. So All I can do now is just go ahead and simplify. Um and let's see what I've got here is when you combine this is going to be 0.5 times this is going to be 65 here And then when you divide this to the other side here, what you're gonna get is you're gonna get negative 20 over like this, except what you also have to do is realize um that there's also another negative sign that's out here. So you can also move this out over here, basically what happens is the negative signs are going to cancel out since you end up with something positive. And that makes sense because we're solving for mass so we shouldn't get a negative number. So we just should get negative M. B. And this is going to equal 1.625 and this is gonna be kilograms. So this is how much boiling wheel water we need. Um it kind of makes sense that we have this number here that was kind of bigger than this number because the final temperature ends up being much much much closer To the 100°C water. And that's just because there's more of it. Anyway, so that's the answer. Guys, let me know if you have any questions