Ice of mass 51.5 g at -10.2° C is added to 218 g of water at 13.5° C in a 111 g glass container of specific heat 0.200 cal/g-°C at an initial temperature of 24.9° C. Find the final temperature of the system.

To find the final temperature of the system, we can use the principle of conservation of energy. The heat lost by the ice and the glass container must be equal to the heat gained by the water.

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

Q = mcΔT

Where:
Q = heat gained or lost in calories
m = mass in grams
c = specific heat capacity in cal/g-°C
ΔT = change in temperature in °C

For the water:
m = 218 g
c = 1.00 cal/g-°C (specific heat of water)

ΔT = final temperature - initial temperature = Tf - 13.5°C

So the heat gained by the water is:
Q1 = (218 g) * (1.00 cal/g-°C) * (Tf - 13.5°C)

Next, let's calculate the heat lost by the ice. The heat lost by the ice can be calculated using the equation:

Q2 = mcΔT

For the ice:
m = 51.5 g
c = 0.50 cal/g-°C (specific heat of ice, assuming it is already at its melting point)

ΔT = final temperature - initial temperature = Tf - (-10.2°C)

So the heat lost by the ice is:
Q2 = (51.5 g) * (0.50 cal/g-°C) * (Tf - (-10.2°C))

Finally, let's calculate the heat lost by the glass container. The heat lost by the glass container can be calculated using the equation:

Q3 = mcΔT

For the glass container:
m = 111 g
c = 0.200 cal/g-°C (specific heat of the glass container)
ΔT = final temperature - initial temperature = Tf - 24.9°C

So the heat lost by the glass container is:
Q3 = (111 g) * (0.200 cal/g-°C) * (Tf - 24.9°C)

According to the principle of conservation of energy, the heat gained by the water must be equal to the sum of the heat lost by the ice and the glass container:

Q1 = Q2 + Q3

Substituting the values we calculated:

(218 g) * (1.00 cal/g-°C) * (Tf - 13.5°C) = (51.5 g) * (0.50 cal/g-°C) * (Tf - (-10.2°C)) + (111 g) * (0.200 cal/g-°C) * (Tf - 24.9°C)

Now, we can solve this equation to find the final temperature Tf. This involves simplifying the equation and solving for Tf. After solving the equation, the final temperature can be calculated.