Suppose 0.0290 kg of steam (at 100.00°C) is added to 0.290 kg of water (initially at 19.7°C.). The water is inside an aluminum cup of mass 44.7 g. The cup is inside a perfectly insulated calorimetry container that prevents heat flow with the outside environment. Find the final temperature (in °C) of the water after equilibrium is reached.

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To solve this problem, we need to apply the principle of conservation of energy. According to this principle, the heat gained by the water and the cup is equal to the heat lost by the steam.

First, let's find the heat gained by the water and the cup. We can use the equation:

Q = mcΔT

Where:
Q is the heat gained (or lost),
m is the mass,
c is the specific heat capacity, and
ΔT is the change in temperature.

The specific heat capacity of water is approximately 4.18 J/g°C, and the specific heat capacity of aluminum is 0.90 J/g°C.

1. Heat gained by the water:
Qwater = mwater * cwater * ΔTwater

Given:
mwater = 0.290 kg
cwater = 4.18 J/g°C
ΔTwater = (final temperature - initial temperature) = (final temperature - 19.7°C)

2. Heat gained by the cup:
Qcup = mcup * ccup * ΔTcup

Given:
mcup = 44.7 g = 0.0447 kg
ccup = 0.90 J/g°C
ΔTcup = (final temperature - initial temperature) = (final temperature - 19.7°C)

The total heat gained by the water and the cup is the sum of these two values:

Q1 = Qwater + Qcup

Now, let's find the heat lost by the steam. We can use the equation:

Q2 = msteam * csteam * ΔTsteam

Given:
msteam = 0.0290 kg
csteam = ?, since we don't have the specific heat capacity of steam
ΔTsteam = (final temperature - initial temperature) = (final temperature - 100.00°C)

We can find the specific heat capacity of steam using a steam table or other reference sources. Typically, the specific heat capacity of steam is around 2.0 J/g°C.

Once we have found the specific heat capacity of steam, we can substitute the given values and calculate Q2.

Since the total heat gained by the water and the cup (Q1) is equal to the heat lost by the steam (Q2), we can set up the equation:

Q1 = Q2

Solving this equation will give us the final temperature (in °C) of the water after equilibrium is reached.