The specific heat of a substance varies with temperature t(°C) as c = 0.20 + 0.14 t + 0.023 t² (cal/gm°C) The heat required to raise the temperature of 2 gm of substance from 5°C to 15°C will be

Correct Answer: 82 calorie

Explanation:

To find the heat required to raise the temperature of a substance when its specific heat capacity varies with temperature, we use the integration method. The small amount of heat $dQ$ required to raise the temperature of mass $m$ by a small temperature range $dt$ is given by:

$dQ=mcdt$

Integrating both sides from the initial temperature $t_1$ to the final temperature $t_2$, we get:

$Q={\int }_{t_1}^{t_2}mcdt$

Given in the problem:
Mass of the substance, $m=2\text{gm}$
Initial temperature, $t_1=5^{\circ }\text{C}$
Final temperature, $t_2={15}^{\circ }\text{C}$
Specific heat capacity function, $c=0.20+0.14t+0.023t^2\text{cal/gm}{°}^{}\text{C}$

Substituting these values into our equation:

$Q={\int }_5^{15}2\left(0.20+0.14t+0.023t^2\right)dt$

$Q=2{\left[0.20t+\frac{0.14t^2}{2}+\frac{0.023t^3}{3}\right]}_5^{15}$

$Q=2{\left[0.20t+0.07t^2+\frac{0.023}{3}{t}^3\right]}_5^{15}$

Now, let us evaluate the expression at the upper limit ($t=15$):

${\text{Value}}_{\text{upper}}=0.20\left(15\right)+0.07\left({15}^2\right)+\frac{0.023}{3}\left({15}^3\right)$

${\text{Value}}_{\text{upper}}=3+0.07\left(225\right)+\frac{0.023}{3}\left(3375\right)$

${\text{Value}}_{\text{upper}}=3+15.75+0.023\left(1125\right)$

${\text{Value}}_{\text{upper}}=3+15.75+25.875=44.625$

Next, evaluate the expression at the lower limit ($t=5$):

${\text{Value}}_{\text{lower}}=0.20\left(5\right)+0.07\left(5^2\right)+\frac{0.023}{3}\left(5^3\right)$

${\text{Value}}_{\text{lower}}=1+0.07\left(25\right)+\frac{0.023}{3}\left(125\right)$

${\text{Value}}_{\text{lower}}=1+1.75+0.9583=3.7083$

Now, substitute these limits back into the calculation for $Q$:

$Q=2\left(44.625-3.7083\right)$

$Q=2\left(40.9167\right)$

$Q\approx 81.83\text{calorie}$

Rounding to the nearest whole number gives:
$Q\approx 82\text{calorie}$

Thus, the heat required to raise the temperature of the substance is 82 calorie.