7. A 200-g block of copper at a temperature of 90°C is dropped into 400 g of water

at 27°C. The water is contained in a 300-g glass container. What is the final
temperature of the mixture?

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To find the final temperature of the mixture, we need to use the principle of conservation of energy and consider the heat gained or lost by each component.

The formula we can use is:

Q = m * c * ΔT

Q: Heat gained or lost
m: Mass
c: Specific heat capacity
ΔT: Change in temperature

Firstly, let's find the heat lost by the copper block.

Qcopper = mcopper * ccopper * ΔTcopper

Given:
mcopper = 200 g
ccopper = 0.39 J/g°C (specific heat capacity of copper)
ΔTcopper = Tf - Tcopper_initial

Next, let's find the heat gained by the water.

Qwater = mwater * cwater * ΔTwater

Given:
mwater = 400 g
cwater = 4.18 J/g°C (specific heat capacity of water)
ΔTwater = Tf - Twater_initial

Now, let's find the heat gained by the glass container.

Qglass = mglass * cglass * ΔTglass

Given:
mglass = 300 g
cglass = 0.84 J/g°C (specific heat capacity of glass)
ΔTglass = Tf - Tglass_initial

Since the system is isolated, the heat lost by the copper block must be equal to the heat gained by the water and glass container.

Qcopper = Qwater + Qglass

Using the formulas above, we can substitute the values and solve for the final temperature (Tf).

Here are the steps:

1. Calculate the heat lost by the copper block:
Qcopper = mcopper * ccopper * ΔTcopper
Qcopper = 200 g * 0.39 J/g°C * (Tf - 90°C)

2. Calculate the heat gained by the water:
Qwater = mwater * cwater * ΔTwater
Qwater = 400 g * 4.18 J/g°C * (Tf - 27°C)

3. Calculate the heat gained by the glass container:
Qglass = mglass * cglass * ΔTglass
Qglass = 300 g * 0.84 J/g°C * (Tf - 27°C)

4. Set up the equation: Qcopper = Qwater + Qglass
200 g * 0.39 J/g°C * (Tf - 90°C) = 400 g * 4.18 J/g°C * (Tf - 27°C) + 300 g * 0.84 J/g°C * (Tf - 27°C)

5. Simplify and solve the equation for Tf (final temperature).

Once you solve the equation, you will find the final temperature (Tf) of the mixture.

To find the final temperature of the mixture, we can use the principle of conservation of energy. The heat lost by the copper block will be equal to the heat gained by the water and the glass container.

The amount of heat lost or gained can be calculated using the formula:

Q = m * c * ΔT

Where:
Q is the heat energy
m is the mass of the object
c is the specific heat capacity
ΔT is the change in temperature

For the copper block, m = 200 g and c = 0.39 J/g°C (specific heat capacity of copper).
The initial temperature of the copper block is 90°C, and we want to find the final temperature.

For the water, m = 400 g and c = 4.18 J/g°C (specific heat capacity of water).
The initial temperature of the water is 27°C, and we want to find the final temperature.

For the glass container, m = 300 g and c = 0.84 J/g°C (specific heat capacity of glass).
The glass container is assumed to be at the same initial temperature as the water, 27°C.

Since the heat gained by the water and the glass container will be equal, we can set up the equation:

(Qcopper block lost) = (Qwater gained) + (Qglass container gained)

Simplifying the equation:

(mcopper block * ccopper block * ΔTcopper block) = [(mwater * cwater * ΔTwater) + (mglass * cglass * ΔTglass)]

Plug in the given values:

(200 g * 0.39 J/g°C * (Tfinal - 90°C)) = [(400 g * 4.18 J/g°C * (Tfinal - 27°C)) + (300 g * 0.84 J/g°C * (Tfinal - 27°C))]

Now, solve for the final temperature (Tfinal).