Solved Problem on Heat
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A body of mass 200 g is heated for 30 seconds by an energy source that provides a power of 210 W at a constant rate. Given the graph of temperature as a function of time, determine the heat capacity of the body, knowing that 1 cal = 4.2 J.


Problem data:
  • Mass of the body:    m = 200 g;
  • Power of the heat source:   \( \mathscr{P} \) = 210 W.
Solution

First, we must convert the power unit given in watts (joules per second) to calories per second (cal/s), in this problem it is more convenient not to use the International System of Units (S.I.)
\[ P=210\;\text{W}=210\;\frac{\text{J}}{\text{s}}=210\;\cancel{\text{J}}\;\frac{1\;\text{cal}}{4,2\;\cancel{\text{J}}}\times \frac{1}{\text{s}}=50\;\frac{\text{cal}}{\text{s}} \]
The heat capacity is given by
\[ \begin{gather} \bbox[#99CCFF,10px] {C=mc} \tag{I} \end{gather} \]
The quantity of heat received (or lost) is given by
\[ \begin{gather} \bbox[#99CCFF,10px] {Q=mc\Delta \theta } \tag{II} \end{gather} \]
where θ was used for temperature in place of t that was used for the time in the problem, substituting the expression (I) into (II), we have
\[ \begin{gather} Q=C \Delta \theta \\ C=\frac{Q}{\Delta \theta} \tag{III} \end{gather} \]
During the heating time, by the graph, we see that the temperature varied from
\[ \begin{gather} \Delta \theta =\theta_{f}-\theta_{i}\\ \Delta \theta =25-15\\ \Delta \theta=10\;\text{°C} \; \tag{IV} \end{gather} \]
The heat received by the body is obtained from the power of the energy source. The source provides 50 calories in 1 second, so in the 30 second heating will provide a quantity Q of heat, using Cross-multiplication, we have
\[ \frac{50\;\text{cal}}{1\;\text{s}}=\frac{Q}{30\;\text{30}} \]
multiplying
\[ \begin{gather} (1\;\text{s})\times (Q)=(30\;\text{s})\times (50\;\text{cal})\\ Q=\frac{30\;\cancel{\text{s}}\times 50\;\text{cal}}{1\;\cancel{\text{s}}} \\ Q=1500\;\text{cal} \tag{V} \end{gather} \]
substituting expressions (IV) and (V) into (III), the heat capacity will be
\[ C=\frac{1500}{10} \]
\[ \bbox[#FFCCCC,10px] {C=150\;\text{cal°C}} \]
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Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .