Question

The specific heat of a substance varies with temperature according to the function $S=0.20+0.14T+0.023T^2$ with $T$ in ${}^{\circ }\text{C}$ and $S$ in ${\text{cal/g}}^{\circ }\text{C}.$ What is the energy required to raise the temperature of $2.0\text{g}$ of this substance from $5^{\circ }\text{C}$ to ${15}^{\circ }\text{C}$?

• A. $50\text{cal}$
• B. $82\text{cal}$
• C. $107\text{cal}$
• D. $230\text{cal}$

Answer 1

The correct option is B $82\text{cal}$

Given that
$S=0.20+0.14T+0.023T^2$
Mass of substance $\left(m\right)=2\text{g}$
Initial temperature $\left(T_i\right)=5^{\circ }\text{C}$
Final temperature $\left(T_f\right)={15}^{\circ }\text{C}$
Now, $dQ=mSdT$
$\Rightarrow \int dQ=\underset{T_i}{\overset{T_f}{\int }}mSdT$

From the data given in the question,
$Q=m\underset{T_i}{\overset{T_f}{\int }}(0.20+0.14T+0.023T^2)dT$
$\Rightarrow Q=2.0\times (0.20T+\frac{0.14T^2}{2}+\frac{0.023T^3}{3}){|}_5^{15}$
$\Rightarrow Q\approx 82\text{cal}$
Hence, option (b) is the correct answer.