How many calories must be added to 10 grams of -20°C ice to warm and completely melt it? Type in your answer as a number below as a number (the units are calories and don't need to be included)

Helpful constants and values (per gram):

Specific Heat of Water: 1 cal/°C
Specific Heat of Ice: 0.5 cal/°C
Specific Heat of Water Vapor: 0.5 cal/°C
Latent Heat of Vaporization: 540 cal
Latent Heat of Fusion: 80 cal

Question

How many calories must be added to 10 grams of -20°C ice to warm and completely melt it? Type in your answer as a number below as a number (the units are calories and don't need to be included)

Helpful constants and values (per gram):

Specific Heat of Water: 1 cal/°C
Specific Heat of Ice: 0.5 cal/°C
Specific Heat of Water Vapor: 0.5 cal/°C 
Latent Heat of Vaporization: 540 cal
Latent Heat of Fusion: 80 cal

BlackTom AI · Accepted Answer

Heat required equals the sum of (1) warming the ice from −20°C to 0°C and (2) the latent heat needed to melt the ice at 0°C.

Relevant knowledge and formulas:
- Sensible heat: q = m · c · ΔT, where m is mass, c is specific heat, ΔT is temperature change.
- Latent heat (fusion): q = m · Lf, where Lf is latent heat of fusion.
Given per-gram values apply; work per 10 g.

Step-by-step calculation with intermediate steps:
1) Warm the ice from −20°C to 0°C:
- m = 10 g, c_ice = 0.5 cal/(g·°C), ΔT = 0 − (−20) = 20°C
- q1 = m · c_ice · ΔT = 10 g · 0.5 cal/g·°C · 20°C
- q1 = 10 · 0.5 · 20 = 100 calories

2) Melt the ice at 0°C (latent heat of fusion):
- Lf = 80 cal/g
- q2 = m · Lf = 10 g · 80 cal/g = 800 calories

3) Total heat required:
- q_total = q1 + q2 = 100 + 800 = 900 calories

Why this is correct:
- The ice must first be raised to its melting point (requires specific-heat energy for the solid phase), then it must undergo a phase change to liquid (requires latent heat). No additional heating of the resulting water or vaporization is required by the problem statement, so only these two contributions are included.

Common mistakes (why other implicit options would be incorrect):
- Using the specific heat of liquid water (1 cal/g·°C) for the ice warming step would overestimate q1.
- Including latent heat of vaporization or heating the liquid above 0°C would add extra energy not asked for.
- Forgetting the ice warming step (only using latent heat) underestimates the required energy.

Conclusion: The total energy required is 900 calories, which matches the provided answer.

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