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

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To raise the temperature of 50.0 g of ice at 0.00°C to 30.0°C, both the latent heat of fusion and the specific heat of water must be considered. The calculation shows that 5500 cal is required, which includes 4000 cal for melting the ice and 1500 cal for heating the resulting water. The book's answer of 1500 cal only accounts for the heating of water, ignoring the melting process. This oversight highlights the importance of considering latent heat in thermal calculations. The discussion emphasizes that without accounting for latent heat, the results can be misleading.
Fantini
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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?
 
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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.
 
I'm guessing this is just an oversight from the book. He even labeled the exercise as Latent Heat. These things happen. :)
 
Fantini said:
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)
 
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