Question

The specific heat of a substance varies with temperature according to the function$\mathit{c}\mathbf{=}\mathbf{0}\mathbf{.20}\mathbf{+}\mathbf{0}\mathbf{.14}\mathit{T}\mathbf{+}\mathbf{}\mathbf{0}\mathbf{.023}{\mathit{T}}^{\mathbf{2}}$ , with T in °C and c in cal/g K. Find the energy required to raise the temperature of $\mathbf{2}\mathbf{.0}\mathbf{}\mathit{g}$of this substance from$\mathbf{5}\mathbf{.0}\mathbf{\text{°C\hspace{0.17em}}}\mathbf{}\mathbf{\text{to\hspace{0.17em}}}\mathbf{}\mathbf{15}\mathbf{\text{°C}}$ .

Short answer

The energy required to raise the temperature of the substance is $82\text{\hspace{0.17em}cal}$

Step-by-step solution

The given data

i) The specific heat of a substance that varies with temperature is$c=0.20+0.14T+0.023T^2$

ii) The mass of the substance is $m=2.0\text{g}$

iii) The initial temperature of the substance is $T_i=5.0\text{°C}$

iv) The final temperature of the substance is $T_f=15\text{°C}$

 Step 2: Understanding the concept of specific heat 

A substance's specific heat capacity is the amount of energy required to raise its temperature by one degree Celsius. We can use the concept of specific heat. By integrating the given function for temperature, we can find the energy.

Formulae:

The heat energy required by a body,$Q=mc\Delta T$ …(i)

Where, $m$= mass

$c$= specific heat capacity

$\Delta T$= change in temperature

$Q$= required heat energy

 Step 3: Calculation of the required heat to raise the temperature

By integrating the expression of specific heat of the substance, that is equation (i) by substituting the value of c, we get the heat required as:

$\begin{array}{c}Q=\underset{T_i}{\overset{T_f}{}}cmdT\\ =m\underset{T_i}{\overset{T_f}{}}0.20+0.14T+0.023T^{2}dT\\ =m{[0.20T+\frac{0.14T^2}{2}+\frac{0.023T^3}{3}]}_{T_i}^{T_f}\\ =(2.0\text{g}){[0.20T+\frac{0.14T^2}{2}+\frac{0.023T^3}{3}]}_{5.0}^{15.0}\\ =(2.0\text{g})[0.20(15.0-5.0)+0.07[{\left(15.0\right)}^2-{\left(5.0\right)}^2]+0.00767[{\left(15.0\right)}^3-{\left(5.0\right)}^3]]\\ =82\text{cal}\end{array}$

Hence, the value of the heat required to raise the temperature is $82\text{cal}$