Temperature and Heat — Question 4

Question 4

Equal heat inputs need not cause equal temperature changes

Two isolated test runs use bodies A and B, each initially at 300 K. Their constant heat capacities are $C_A=200\,\mathrm{J/K}$ and $C_B=500\,\mathrm{J/K}$. With no work or phase change, use $Q=C(T_f-T_i)$. Each receives 2.00 kJ.

1. Find both final temperatures and the ratio of their rises.

2. How much heat must each receive to reach 315 K? Explain why equal mass alone would not settle either comparison.

Question 4 - Solution

Equal heat inputs need not cause equal temperature changes

(a) Method and calculation

$$\Delta T_A=2000/200=10.0\,\mathrm K,\quad\Delta T_B=2000/500=4.00\,\mathrm K.$$$$T_{f,A}=310\,\mathrm K,\quad T_{f,B}=304\,\mathrm K,\quad\frac{\Delta T_A}{\Delta T_B}=2.50.$$

(b) Interpretation and further calculation

$$Q_A=200(315-300)=3000\,\mathrm J.$$$$Q_B=500(315-300)=7500\,\mathrm J.$$Heat capacity, not mass alone, fixes the response. Even at equal mass, differing material-specific capacities can give unequal $C$.

Physical check

For the same positive heat input, the larger heat capacity produces the smaller rise.