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A $44.0$ -$\mathrm{g}$ sample of an unknown metal at $99.0^{\circ} \mathrm{C}$ was placed in a constant-pressure calorimeter containing $80.0 \mathrm{g}$ of water at $24.0^{\circ} \mathrm{C} .$ The final temperature of the system was found to be $28.4^{\circ} \mathrm{C}$. Calculate the specific heat of the metal. (The heat capacity of the calorimeter is $12.4 \mathrm{J} /^{\circ} \mathrm{C}$ )

$s_{M}=0.491 J /^{\circ} \mathrm{C} \cdot g$

Chemistry 101

Chapter 6

Thermochemistry

Carleton College

University of Maryland - University College

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So in this problem, we know that some unknown metal is placed in a constant pressure Keller emitter containing water and that after this medalist placed, it reaches an equilibrium temperature of 28.4 degrees Celsius. So we'll just write that as the temperature final. So we know that the fundamental way that we're going to be approaching this problem is that the amount of heat that the metal gives off is the amount of heat that the water absorbs. However, there is one catch here. We do know that we cannot ignore the heat that the Keller emitter absorbs, which is why the problem gives us a specific key capacity of the clock Keller emitter to be 12 point for Jules per degree Celsius. So instead, our approach is gonna have to include and take that into account. So instead, we're going to write that the negative change in temperature or the changing heat. In other words, the amount of heat that the metal gives off key to the metal is going to equal to the change in heat of the water, plus the changing heat of the calorie meter. So from here we can just plug in our values and take advantage of this specific equation that cure heat is equal to mass kind of specific heat capacity. Times change in temperature foots. First do the left side were finding the changing sheet of metal, so we know that the mass of the metal it's 44 point no grams, which is given to us in the problem. We don't know the specific keep capacity. That's what we're trying to solve. And we do know that the change in temperature can be calculated. View the initial and final temperatures. So we know that the initial temperature of the metal was 99 degrees Celsius in the final was 20.4 degree Celsius. So we can write the change in temperature and such, so that gives us expression for the changing heat of the metal. Now what do you want to do is to the exact same thing for the water and the calorie murder. So for the water, So I'll just put this as the metal for the water. We know that the mass of the water was 80 of 80 grams because that's how much that was given to us in the problems we could just great 80 grams. You know that the specific heat capacity of water It's 4.14 rules per gram for Celsius. We also know that the initial temperature of the water was 24 degrees Celsius and final temperature was 20.4. So it got 4.4 degrees Celsius harder. So that's the H 20 And finally, we wanted to do the Keller emitter. So we know that the calorie murder doesn't have a mass, so we can just ignore that. But instead you do it. No, it's specific heat capacity, 12.4 Jules degree centigrade, and we know that it's the exact same as water. So the reason as to why you know that you don't need the mass of the Calumet er is because the heat capacity that were given doesn't have a grams the denominator, Which is why we know that we won't have to use unless because it's independent of mass. That is for the calorie meter. So now we can rewrite this expression at the following when we calculate all of this, we get that the specific heat of the metal is we're gonna rearrange this. It's gonna be 80 grams, 4.184 Plus, this is a kalamata. Now we're moving on to, and now we have to divide some constant from the metal. So I'll have to do is divide in negative. 44 men divide this value. So all that I've just done is I've isolated the specific heat capacity from this initial equation from this overall formula. And all you need to do now is you plug it into your calculator and that you'll get that the specific heat capacity of this metal. 0.4 91 Jules for Grant sent agreed, and that is the final answer.

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