It is a well-­--known fact that Dr. HerB is extremely fond of coffee. However, f

Question

It is a well-­--known fact that Dr. HerB is extremely fond of coffee. However, freshly brewed
coffee can be perilously hot, so Dr HerB likes to put an ice cube or two in his coffee in order to
bring it down to a more palatable temperature for immediate consumption.
In this problem you will determine the volume of ice Vi, initially at a temperature Ti needed to
bring down the temperature of a volume of coffee Vc from its initial temperature Tc to a more
pleasant final temperature Tf. (Assume that the coffee and ice are thermally insulated from
their surroundings, and that coffee has the same specific heat and density as water)
(b) The ice gains heat in three steps. Denote these three steps by Q1, Q2 and Q3 and explain what each step consists of.
(c) Express Qtot = Q1 +Q2 +Q3 in terms of the variables given and relevant physical constants.
(d) Give an expression for the heat Qc lost by the coffee.
(e) Obtain an expression for Vi.

Explanation / Answer

b) Q1=heat gained by ice in heating up to melting point 273 K= iViCp,i(273-Ti)

=>Q1=iViCp,i(273-Ti)

Q2=latent heat gained by ice in melting from 273 K to water at 273 K=iViLf

=>Q2=iViLf

Q3=heat gained by the water melt from ice from 273 K to final temperature Tf=iViCp,w(Tf-273)

=>Q3=iViCp,w(Tf-273)

c)Qtot=Q1+Q2+Q3

=>Qtot=iVi[Cp,i(273-Ti)+Lf+Cp,w(Tf-273)]

d)Qc=wVwCp,w(Tc-Tf)

e)

from heat balance Qtot=Qc

=>iVi[Cp,i(273-Ti)+Lf+Cp,w(Tf-273)]=wVwCp,w(Tc-Tf)

=>Vi=wVwCp,w(Tc-Tf)/[i{Cp,i(273-Ti)+Lf+Cp,w(Tf-273)}]

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