Help with a conservation of heat problem

[discordplus]

I already did it but its a different problem type that I really was unsure about so I just want to verify my solution.

Homework Statement

Problem: A 350-gram piece of metal at 100 C is dropped into a 100-gram aluminum cup containing 500 g of water at 15 C. The final temperature of the system is 40 C. What is the specific heat of the metal, assuming no heat is exchanged with the surroundings? Specific heat for aluminum is around 900 J/kg*K.

Basically:
m, m = .35kg
m, w = .5kg
m, a = .1kg
Ti, m = 100 C
Ti, a,w = 15 C
Tf = 40 C
Cp,w = 4186 J/kg*K
Cp,a = 900 J/kg*K
Cp,m = ?

Homework Equations

Q,a = -Q,b
delta Q = m*Cp*delta T

The Attempt at a Solution

how i did it:
Q,w,a = -Q,m
((4186 J/kg*K)(40-15)(.5) + (.1)(40-15)(900)) = -(.350)(C,m)(40-100)
52250 J + 2250 J = 21C,m
54500 = 21C,m
C,m = 2595.238 J/kg*K

Alright, step one here is really what I'm unsure about. I treated the aluminum and water as one part of the system, therefore I added the heat gained by the water and the aluminum as shown above, which I wasn't entirely sure of (in terms of that being allowed). Also I know a change in absolute temperature in K is the same as a change in temperature, deg C, but I was shaky on the interchangable use of kelvins and celsius degrees in the specific heats. I'm pretty sure that's valid, but I just would like to have that reaffirmed.

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Anyways, that corresponded pretty closely with one of the choices, 2600 J/kg*deg K so I'm just making sure the work is right because usually I'm only working with two objects in the system, not three. Thanks in advance.

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Also: Can conductive heat transfer only occur when a solid can mediate the energy transfer? I think this is true, but I'm not sure. I'm loose about the methods of heat transfer, very loose.

 

[LowlyPion]

Welcome to PF.

Your method looks OK.

ΔT in °K = ΔT in °C

When the heat equalizes you know how much the one cooled down by the amount the others heated.

 

[nlightner]

Hello,

First of all, your method for solving the problem seems correct. You correctly identified the different masses and specific heats of the metal, water, and aluminum and used the appropriate equations to solve for the specific heat of the metal. As for your first question, it is perfectly valid to treat the water and aluminum as one part of the system and add their heat gains together. This is because they are both at the same temperature and will experience the same change in temperature, so their heat gains can be added together.

Regarding the use of kelvins and degrees Celsius, it is important to note that the specific heat values are defined in terms of kelvins (J/kg*K). However, when dealing with temperature changes, the change in temperature (delta T) can be expressed in either kelvins or degrees Celsius since they have the same magnitude. So in your calculation, it is valid to use 40-15=25 degrees Celsius or 40 K-15 K=25 K. It is also important to note that the specific heat values may vary slightly depending on the units used (J/g*K or J/kg*K), so it is always good to double check the units and make sure they are consistent.

To answer your last question, conductive heat transfer can occur between any two objects as long as they are in contact with each other. It does not necessarily require a solid to mediate the energy transfer, but solids are typically better conductors of heat compared to liquids or gases. However, in this specific problem, the heat transfer is occurring through conduction between the metal and the water/aluminum, so the metal acts as the mediator in this case.

Overall, your solution seems correct and your understanding of the concepts is good. Keep up the good work!