# Example Problem with Complete Solution

5C-10 : Thermal Equilibration of a Copper Block with an Iron Block 5 pts
When a block of cold carbon steel touches a hot aluminum block in an insulated chamber, both blocks eventually reach thermal equilibrium at a temperature between their initial temperatures.
Determine the equilibrium temperature if the aluminum block has a mass of 10 kg and an initial temperature of 380oC and the carbon steel block has a mass of 25 kg and an initial temperature of 10oC.

Read : The mass and initial temperature of each block are given.  We know that the equilibrium temperature of the blocks must lie between the two intial temperatures.  We will need to lookup the heat capacity or specific heat of both steel and aluminum.  Then, we can apply the 1st Law to a system made up of the two blocks.  No work or heat transfer crosses the boundary of this system during the approach to equilibrium, so the only unknown in the equation is the final, equilibrium temperature.  So, we can solve for it and evaluate it.
Diagram: A diagram is optional here. The system and process are fairly simple.
Given: msteel 25 kg mAl 10 kg
Tsteel,1 10 oC TAl,1 380 oC
Find: T2 ??? oC
Assumptions: 1 - Steel and aluminum have constant heat capacities.
2 - No heat is exchanged with the surroundings by either the steel or the aluminum.
3 - Steel and aluminum are both incompressible, so this process is a constant volume process.
Equations / Data / Solve:
We begin by writing the 1st Law and we choose as our system the steel and the aluminum.
Eqn 1
By cleverly selecting our system, Q = 0 and W = 0.  This makes the solution simpler.
Because both steel and aluminum are assumed to be incompressible with constant heat capacities: Eqn 2
Substitute Eqn 2 into Eqn 1 twice, once for steel and once for aluminum to get :
Eqn 3
Notice that there is only one T2 because in the final, equilibrium state, the steel and the aluminum are both at the same temperature !
Now, solve Eqn 3 for T2:
Eqn 4
Now, solve Eqn 4 for T2.
Eqn 5
Before we can evaluate T2, we must lookup the specific heat of steel and of aluminum.
For carbon steel, I found : CP,st 0.49 kJ/kg-K
For incompressible solids, CV = CP, so : CV,st 0.49 kJ/kg-K
For aluminum, I found : CP,Al 0.91 kJ/kg-K
CV,Al 0.91 kJ/kg-K
Now, we can finally plug numbers into Eqn 5 and evaluate T2: T2 167.7 oC
Verify: None of the assumptions can be verified.