Question

a 155 gram sample of copper was heated to 150.0ºC then placed into 250 gram of water at 19.8ºC. (The specific heat of copper is 0.385J/gºC.)Calculate the final temperature of the mixture. (assume no heat loss to the surroundings)

Answer 1

What I know: Copper:155 g
Water:250.0 g
t: 150ºC-19.8ºC=130.2ºC
How do I set up?

Answer 2

You want to set it up such that Heat Gained = -Heat Lost. Have you heard of this or do you want me to show you a little more?

Answer 3

Show me a little more please.

Answer 4

Sure. I'll ask you a series of questions so you hopefully understand the process. Which is gaining heat and which is losing heat? Copper or Water & how do you know?

Answer 5

Water is gaining heat and copper is losing heat

Answer 6

Sounds good. You know this because copper is hotter than water. Have you seen the "Heat Gained = - Heat Lost" equation before?

Answer 7

no

Answer 8

Have you heard of calorimetry?

Answer 9

Yes I have.
Would the formula I have to use be
heat gained by water= (mass)(specific heat)( ∆t)?

Answer 10

Yes but you use it twice. Let me explain. You have 2 things. One is clearly hot and the other is cold. in an isolated system (meaning those 2 things are the only things that interact with one another) the thermal energy is going to be exchanged only between those two objects and nothing else. This is an ideal case.

You told me that the water will gain heat and the copper will lose heat. This being the case, the total thermal energy between the two objects does not change (according to the conservation of energy - energy is never created or destroyed, only transferred from one form to another). So if we set this up in the form of an equation we can see:
\[Q _{Lost} + Q _{Gained} = 0\]
\[Q _{Gained} = -Q _{Lost}\]
But because the copper is losing and the water is gaining:
\[Q _{Water} = -Q _{Copper}\]
I will show you how to use the mC Delta t stuff next. Do you have any questions so far?

Answer 11

Okay sounds good. I understand so far.

Answer 12

Ok. Now lets talk about your mass, specific heat and temp change by defining the variables.
Q stands for heat and has units of Joules (J)
m stands for mass and has units of (g)
C stands for specific heat and has units of (J/(g deg C))
Tf is the final temperature of a substance (deg C)
Ti is the initial temperature of a substance (deg C)

The equation for a given heat calculation is:
\[Q = mC \Delta T\]
Which is also written as
\[Q = mC(T _{f}-T _{i})\]

Still ok?

Answer 13

Yes

Answer 14

Ok. What we are going to do is make a list of what we know and what we are trying to find. I want you to tell me the m, C, Ti, and Tf for both copper and water. Tell me what we are trying to find out of those variables as well. If you dont know something or have a number that you think you should, let me know and we'll talk about it. Then, once this is taken care of, we will set up our master equation that looks like an algebra problem to solve and its pretty straight forward from there.

Answer 15

m (copper): 155g
m (water): 250.0g
C: 0.385 J/gºC
Ti (copper): 150.0ºC (not sure if this is consider ti)
Ti (water): 19.8ºC (not sure also)
Tf: We are trying to solve for this. The final temperature of the mixture.

Answer 16

I think Ti would be 130.2ºC

Answer 17

I like what I see above for your initial conditions but I do have a few questions for you. What is C for water? What do you mean Ti would be 130.2? (How do you do degree symbol? I'm curious.)

Answer 18

I do not know what C is for water. For Ti, what I did was 150.0ºC-19,8ºC but i guess that is wrong. So i am assuming what I wrote above is correct:
Ti (copper): 150.0ºC (not sure if this is consider ti)
Ti (water): 19.8ºC (not sure also)
(For the degree symbol you hold alt and click on zero. º)

Answer 19

Thanks for the tip.

Answer 20

You welcome.

Answer 21

The C for water is not provided. This is a standardized value that can be found in the internet. Typically C for water is 4.816 J/(gC). We will use this number. Do not subtract any numbers or do anything at this point for Ti. We are just seeing what we have.

What is the problem asking us to find?

Answer 22

Okay.
The problem is asking for the final temperature of the mixture.

Answer 23

Yes you are correct. (sorry pc started to get slow on me...)
We will call Tf = x. We do not know this for either material. This is the point where we set up our algebra problem as shown below.
\[Q _{Water} = -Q _{Copper}\]
\[m _{w}C _{w}(T _{f}-T _{i})_{w} = -m _{cu}C _{cu}(T _{f}-T _{i})_{cu}\]
That make sense so far?

Answer 24

It's okay.
Yes makes sense.

Answer 25

Ok. Now we plug in numbers.
\[(250)(4.186)(x-19.8)=-(155)(0.385)(x-150)\]
Can you reduce this down on both sides and tell me what you get before solving for x?

Answer 26

Would i put my answers to the correct significant figure?

Answer 27

Yes please.

Answer 28

(1047x- 2.07x10^4)=59.675x-8.95x10^3
I do not think I wrote it to the correct significant figure.

Answer 29

Oh I'm sorry, I thought you were giving me an answer. Typically, I never go to correct sig figs until the very end. This is very close to what I have. Let me show you what I have.
\[1046.5x -20720.7 = -59.675x + 8951.25\]
Sound good so far?

Answer 30

Can you rearrange now and solve for x?

Answer 31

Okay I see.
1106.175x=29671.95

Answer 32

keep going.

Answer 33

x=26.82392027
x=2.682?

Answer 34

Before sig figs, I agree with first one x = 26.82392027

Answer 35

I see 3 sf in all of the variables. I do not consider constants such as specific heat of water in my sf calculation because it is known to be exactly what the number is unless otherwise indicated. Mass for both quantities have 3 sf (assuming it is 250g exactly). Same goes for Ti, 3sf as long as it is 150 exactly.

Answer 36

Wait I have a question.
You mentioned water is 4.816 J/(gºC) but in the equation you wrote water is 4.186.
Which is the correct one?

Answer 37

4.186... sorry dyslexia kicks in

Answer 38

But i still get the same answer as you did despite that minor difference. I did a google search to verify that number.

Answer 39

So what do you think the final answer will be with correct sf?

Answer 40

2.68?

Answer 41

You mean 26.8?

Answer 42

Oh yes, 26.8

Answer 43

That would be the best answer ever...

Answer 44

Does this stuff make more sense to you now?

Answer 45

Yes much more sense!
I was mainly having trouble creating the set up but now I understand. Thanks

Answer 46

Not a problem at all. Glad I could be of help.

Answer 47

Just one last question. Would it be 26.8J or no J?

Answer 48

The correct units would be degrees Celsius. This is because we are talking about temperature. Heat (Q) has units of J (joules)

Answer 49

Heat is energy, temperature is a relative measure of that energy so they are different things.

Answer 50

Okay makes sense thanks again!

Answer 51

No problem. Later.

Answer 52

Later.