Question

The specific heat of a substance at temperature ${t^ \circ }C$ is $s = a{t^2} + bt + c$. The amount of heat required to raise the temperature of $m$ grams of a substance from ${0^ \circ }C$ to ${t^ \circ }C$.

Answer 1

Specific heat capacity $\left( s \right)$ is the amount of heat required to raise the temperature of unit mass of the substance through ${1^ \circ }C$ or $\left( {1K} \right)$ and heat capacity$\left( {{C^|}} \right)$ is the amount of heat required to raise the temperature of the whole body by ${1^ \circ }C$.

Complete step by step answer:
Given: $s = a{t^2} + bt + c$ $............\left( 1 \right)$
Mass$= m$ and temperature of the substance changes from ${0^ \circ }C$ to ${t^ \circ }C$.
Let $q$ be the amount of heat required to raise the temperature of $m$ grams of substances from ${0^ \circ }C$ to ${t^ \circ }C$.
We know that, heat in terms of specific heat capacity can be given as
$q = \int\limits_0^t {msdt}$ $............\left( 2 \right)$
Amount of heat required $q = \int\limits_0^t {msdt}$
Substituting equation $\left( 1 \right)$ in equation$\left( 2 \right)$, we get
$\Rightarrow q = \int\limits_0^t {m\left( {a{t^2}bt + c} \right)} dt$ ($\because$Temperature of the substance changes from ${0^ \circ }C$ to ${t^ \circ }C$)
On integrating with respect to $t$ , the above equation becomes
$\Rightarrow q = m \times \left( {\dfrac{a}{3}{t^3} + \dfrac{b}{2}{t^2} + ct} \right)_0^t$ $\left( {\because \int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}}} } \right)$
On applying the limits, we get
$\Rightarrow q = m \times \left[ {\dfrac{a}{3}\left( {{t^3} - 0} \right) + \dfrac{b}{2}\left( {{t^2} - 0} \right) + c\left( {t - 0} \right)} \right]$
On simplifying the above equation
$\therefore q = m\left( {\dfrac{{a{t^3}}}{3} + \dfrac{{b{t^2}}}{2} + ct} \right)$

Therefore, heat required to raise the temperature of mass m from ${0^ \circ }C$ to ${t^ \circ }C$ is $q = m\left( {\dfrac{{a{t^3}}}{3} + \dfrac{{b{t^2}}}{2} + ct} \right)$.

Note: Heat is a form of internal energy which is obtained due attractive force of molecules present in a body and random movement of molecules whereas temperature is the quantity which is necessary to determine the direction in which heat flows when two bodies are kept in contact. The $S.I$ unit of heat is joules$\left( J \right)$ and the $S.I$unit of temperature is Kelvin $\left( K \right)$.