To find the final equilibrium temperature when two objects exchange heat, you can use the principle of heat transfer. In this case, both bricks are made of the same material and have different volumes. You can use the concept of heat transfer through conduction.
The heat transfer equation for conduction is given by:
�
=
�
�
\DeltaT
Q=mc\DeltaT
Where:
�
Q is the heat transferred.
�
m is the mass of the object.
�
c is the specific heat capacity of the material.
Δ
�
ΔT is the change in temperature.
Since the bricks are made of the same material, their specific heat capacity (
�
c) is the same.
Let's consider the two bricks separately:
For Brick 1:
Volume is half that of Brick 2.
Mass is directly proportional to volume since the material is the same.
For Brick 2:
Volume is twice that of Brick 1.
Mass is directly proportional to volume since the material is the same.
Now, let's consider the heat transfer for each brick:
Brick 1:
�
1
=
�
1
�
\DeltaT
1
Q
1
=m
1
c\DeltaT
1
�
1
=
(
0.5
�
2
)
�
\DeltaT
1
Q
1
=(0.5m
2
)c\DeltaT
1
Brick 2:
�
2
=
�
2
�
\DeltaT
2
Q
2
=m
2
c\DeltaT
2
Since heat is conserved, the heat lost by one brick is gained by the other. Therefore:
�
1
=
−
�
2
Q
1
=−Q
2
Substituting the expressions for
�
1
Q
1
and
�
2
Q
2
:
(
0.5
�
2
)
�
\DeltaT
1
=
−
�
2
�
\DeltaT
2
(0.5m
2
)c\DeltaT
1
=−m
2
c\DeltaT
2
Now, you can calculate the final equilibrium temperature. Let
�
�
T
f
be the final temperature:
0.5
�
2
�
(
�
�
−
30
)
=
−
�
2
�
(
�
�
−
60
)
0.5m
2
c(T
f
−30)=−m
2
c(T
f
−60)
Simplify and solve for
�
�
T
f
:
0.5
(
�
�
−
30
)
=
−
(
�
�
−
60
)
0.5(T
f
−30)=−(T
f
−60)
Now, solve for
�
�
T
f
:
0.5
�
�
−
15
=
−
�
�
+
60
0.5T
f
−15=−T
f
+60
Add
�
�
T
f
to both sides:
0.5
�
�
+
�
�
=
60
+
15
0.5T
f
+T
f
=60+15
1.5
�
�
=
75
1.5T
f
=75
Now, divide by 1.5:
�
�
=
50
T
f
=50
So, the final equilibrium temperature for both bricks will be
50
°
�
50°C.