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202212231640 Solution to 2016-DSE-PHY-IA-2

0.3\,\mathrm{kg} of water at temperature 50\,^\circ\mathrm{C} is mixed with 0.2\,\mathrm{kg} of ice at temperature 0\,^\circ\mathrm{C} in an insulated container of negligible heat capacity. What is the final temperature of the mixture?

Given: specific heat capacity of water =4200\,\mathrm{J\,kg^{-1}\,^\circ C^{-1}}; specific latent heat of fusion of ice 3.34\times 10^5\,\mathrm{J\, kg^{-1}}.

Roughwork.

For all 0.3\,\mathrm{kg} liquid water, a temperature drop of 50\,\mathrm{C^\circ} will release

\begin{aligned} \textrm{Sensible heat} & = (0.3)(4200)(50) \\ & = \textrm{63,000}\,\mathrm{J} \\ \end{aligned}

and a phase transition of freezing will release further latent heat (of fusion)

\begin{aligned} \textrm{Latent heat}& = ml_f \\ & = (0.3)(3.34\times 10^5) \\ & = \textrm{100,200}\,\mathrm{J} \\ \end{aligned}

whereas for 0.2\,\mathrm{kg} of solid ice to melt all at once, latent heat (of liquidization)

\begin{aligned} \textrm{Latent heat} & = ml_f \\ & = (0.2)(3.34\times 10^5) \\ & = \textrm{66,800}\,\mathrm{J} \\ \end{aligned}

is to be absorbed.

Lemma.

Heat always moves from hotter objects to colder objects, unless energy in some form is supplied to reverse the direction of heat flow.

Wikipedia on Second law of thermodynamics

By the inequalities

0<\textrm{66,800}-\textrm{63,000}<\textrm{100,200}

one will know.