How Much Heat Is Needed to Melt Ice and Warm It to 30°C?

[Fantini]

Here's the problem.

How much heat is required to raise the temperature of $50.0$g of H$_2$O ice at $0.00^{\circ}$C to $30.0^{\circ}$C? Assume an average $1.00 \text{ cal/g}^{\circ}$C specific heat for water in this temperature range.

Since he said ice I assumed all $50.0$g is ice and therefore you need to both convert the ice to water than then heat the water. This amounts to

$$Q = mL + mc \Delta T = 50 \cdot 80 + 50 \cdot 1 \cdot 30 = 5500 \text{ cal}.$$

However the book answer is $1500$ cal, which I disagree. This is the amount of heat necessary to heat the water, but not to turn all ice to water.

Am I wrong?

 

[Ackbach]

Well, the book is not converting the ice to water, right? If you just do the $m c \Delta T$ bit, you get the book's answer.

I would definitely have approached it the way you did, because I'm not sure I can conceive of a block of ice at $30^{\circ}\text{C}$. I suppose you could prove the book wrong if, in a vacuum (best-case scenario), heating the block of ice to $30^{\circ}\text{C}$ would have to melt it. It is true that the heat required to melt is considerably more than the heat required to raise the temperature. Still, my intuition is strongly on your side.

 

[Fantini]

I'm guessing this is just an oversight from the book. He even labeled the exercise as Latent Heat. These things happen. :)

 

[I like Serena]

I'm guessing this is just an oversight from the book. He even labeled the exercise as Latent Heat. These things happen. :)

So it looks like an example what would happen if you do not take latent heat into account. (Wink)

 

[CellCoree]

I would like to clarify that your approach is correct. The book answer seems to be incorrect as it only takes into account the heat required to raise the temperature of the water, but not the heat required to melt the ice first. The correct answer should be 5500 cal, as you have calculated. It is important to consider both the heat of fusion and the specific heat when calculating the total heat required to change ice to water and then raise its temperature.