Michael S. answered • 13d
B.S. in Chemistry, Indiana University; Organic Chem Teaching Intern
q = 31.1 kJ
Two distinct processes happen here, so the answer is a sum of two terms. Sketching a heating curve first makes this obvious: a sloped line while the solid warms, then a flat plateau while it melts.
Step 1 — warm the solid to its melting point
q1 = mcΔT
ΔT = 327.5 − 23.0 = 304.5 °C
q1 = (500.0 g)(0.128 J/g·°C)(304.5 °C) = 19 488 J = 19.49 kJ
Step 2 — melt it at 327.5 °C
ΔHfus is given per mole, so convert the mass first — this is the step people skip:
n = 500.0 g ÷ 207.2 g/mol = 2.413 mol
q2 = (2.413 mol)(4.80 kJ/mol) = 11.58 kJ
Step 3 — add them
qtotal = 19.49 + 11.58 = 31.1 kJ
The temperature does not rise during melting
That flat plateau on the heating curve is the conceptual heart of the problem. All 11.58 kJ of the second step goes into breaking the metallic lattice apart, not into raising the temperature — the lead sits at 327.5 °C the whole time it is melting. That is why you cannot handle this with a single mcΔT; there is no ΔT during a phase change.
Watch the units
Specific heat is in J/g·°C while enthalpy of fusion is in kJ/mol. Adding 19 488 to 11.58 without converting gives a meaningless number, and it is the most common error on this problem. Pick one unit and convert everything to it before adding.
Sanity check on the split: heating accounts for about 63% of the energy and melting about 37%. That is reasonable given the large 304.5 °C climb against lead's fairly small heat of fusion. If your melting term had dominated, it would be worth rechecking the mole conversion.
Extending it: if the problem had asked you to also raise the resulting liquid to some higher temperature, you would add a third term using the liquid specific heat, which differs from the solid value. And running the whole thing backwards — cooling and freezing — gives the same magnitude with a negative sign, since heat is released rather than absorbed.
