Stir 0.25 kg of ice, initially at - 5oC, into 0.5 kg of tea, initially at 45oC. What is the final temperature?
cw = 4186 J/(kg Co)

ci = 2090 J/(kg Co)

Lf = 3.33 x 105 J/kg

It seems reasonable to expect all the ice to melt and to have a final temperature between 0oC and 45oC. We will simply call this final temperature Tf. Then the change in temperature for the tea is

T1 = Tf - T1i

Anything = change in anything = (final value) - (initial value)

T1 = Tf - 45oC

The heat transferred from the tea (or by the tea ) is

Q1 = cw m1 T1

Q1 = [4186 J/(kg Co)] [0.5 kg] [Tf - 45oC]

Notice that [Tf - 95oC] will be negative which means Q1 will be negative. That means that Q1 represents heat flowing from or out of the tea.

Q1 = [4186 J/(kg Co)] [0.5 kg] [Tf - 45oC]

Q1 = 2093 Tf J - 94,185 J

I have dropped the Co in the first term, requiring that we measure Tf in units of Co.

What about the change in temperature for the ice? What about the heat absorbed by the ice?

First the ice changes temperature from - 5oC to 0oC or a change of T2a = 5oC -- as ice! That involves heat of

Q2a = c2 m2 T2a

Q2a = [2090 J/(kg Co)] [0.25 kg] [0oC - ( - 5oC)]

Q2a = [2090 J/(kg Co)] [0.25 kg] [5oC]

Q2a = 2612.5 J

This is the heat required to raise the ice to a temperature of 0oC.

Now all of the ice melts. That requires heat of

Q2b = m2 Lf

Q2b = [0.25 kg] [3.33 x 105 J/kg]

Q2b = 8.33 x 104 J = 83,300 J

Now, this 0.25 kg of water at 0oC is raised in temperature to the final temperature Tf; that requires heat of

Q2c = cw m2 T2

Q2c = [4186 J/(kg Co)] [0.25 kg] [Tf - 0oC]

Q2c = [1046.5 J/Co] [Tf ]

Thus, the total heat absorbed by the ice is

Q2 = Q2a + Q2b + Q2c

Q2 = 2,612.5 J + 83,300 J + 1046.5 Tf J

Again, I have dropped the Co in the last term, requiring that we measure Tf in units of Co.

Now we can apply the Work-Energy Theorem or say that energy is conserved. This means

Q1 + Q2 = 0

[2093 Tf J - 94,185 J] + [2,612.5 J + 83,300 J + 1046.5 Tf J] = 0

[2093 Tf - 94,185 ] + [2,612.5 + 83,300 + 1046.5 Tf ] = 0

3139.5 Tf - 8,272.5 = 0

3139.5 Tf = 8,272.5

Tf = 8,272.5 /3139.5

Tf = 2.6

Of course, this means

Tf = 2.6oC