Question

A piece of ice (Heat capacity = $2100Jk{g^{ - 1}}{}^ \circ C$ and latent heat =$3.36 \times {10^5}Jk{g^{ - 1}}$) of mass m grams is at $ - {5^ \circ }C$ at atmospheric pressure. It gives 420 J of heat so the ice starts melting. Finally, when the ice-water mixture is in equilibrium, it is found that 1gm of ice has melted. Assume that there is no other heat exchange in the process, the value of m is-
A) 8
B) 7
C) 9
D) 6

Answer 1

Hint: The total heat given to the ice is utilized in two ways some heat us used as latent heat and rest used as specific heat to raise the temperature. Hence total heat,
$Q = $Latent heat + specific heat
$Q = mL + mc\Delta T$
Where $m$= mass of substance, $L$=latent heat, $c$= specific heat capacity and = rise in temperature

Complete step by step answer:
Step1: As given that heat capacity c=$2100Jk{g^{ - 1}}{}^ \circ C$
Latent heat =L=$3.36 \times {10^5}Jk{g^{ - 1}}$
Now as heat is absorbed at ${0^ \circ }C$ Therefore $\Delta T = 0 - \left( { - 5} \right) = {5^ \circ }C$
Total heat given to ice $Q = 420J$
Let the mass of the ice = $m$ =?
Since the total heat given to the ice is utilized in two ways some heat is used as latent heat and rest is used as specific heat to raise the temperature. Hence total heat,
$Q = $Latent heat + specific heat
$Q = mL + mc\Delta T$ …………….(1)
Where m= mass of substance, $L$=latent heat, $c$= specific heat capacity and = rise in temperature
Step2: substitute all the value in equation (1) and calculate $m$,
\[
  420 = \left( {1 \times 3.36 \times {{10}^5} + m \times 2100 \times 5} \right) \times {10^{ - 3}} \\
   \Rightarrow m = 8g \\
 \]

$\therefore $ The value of m is 8g. Hence option (A) is the correct answer.

Additional information:
The specific heat capacity, of a substance, is the heat capacity of a sample of the substance divided by the mass of the sample. It is the amount of energy that must be added, in the form of heat, to one unit of mass of the substance in order to cause an increase of one unit in its temperature.
It is given by-
$Q = mc\Delta T$
Where m=mass of body, c=heat capacity and
$\Delta T$=increase in temperature.

Note:
Keep in mind that the latent heat is defined as the amount of heat required to change the phase of a substance from a solid to liquid or liquid to gas and vice-versa. If the body directly converts from solid to gas then this heat is called the heat of sublimation.