The Absorption of Heat by Solids and Liquids — Question 10

Question 10

An energy budget sets the final phase as well as temperature

A sealed idealized sample contains .200 kg of a substance with melting temperature 300 K. Initially it is entirely solid at 290 K. Take $c_s=1000$, $c_l=2000$ J/(kg K), and $L_f=50000$ J/kg, all constant. Add heat with no losses or work; no other phase transition is accessible.

1. For a total input of 8.00 kJ, determine final temperature and liquid mass fraction.

2. Construct $T(Q)$ for nonnegative input through the solid, mixed-phase and liquid ranges. Evaluate the temperature for 16.0 kJ.

Question 10 - Solution

An energy budget sets the final phase as well as temperature

(a) Method and calculation

$$Q_1=mc_s(300-290)=2000\,\mathrm J,\quad Q_2=Q_1+mL_f=12000\,\mathrm J.$$$$2000<8000<12000\ \Rightarrow\ T_f=300\,\mathrm K,\quad f_l=\frac{8000-2000}{10000}=.600.$$

(b) Interpretation and further calculation

With $Q$ in joules,$$T(Q)=\begin{cases}290+Q/200,&0\le Q\le2000,\\300,&2000\le Q\le12000,\\300+(Q-12000)/400,&Q\ge12000.\end{cases}$$$$T(16000)=300+4000/400=310\,\mathrm K.$$Both endpoints are continuous, while the slope changes at each phase boundary.

Physical check

The mixed-phase branch is an interval of energy, not a jump in temperature.