91Ó°ÊÓ

1. The mass of the sample is $\mathrm{m}=0.400\mathrm{kg}$
2. The time of the sample is${\mathrm{t}}_{\mathrm{s}}=80.0\min$
3. The specific heat of the sample is $\mathrm{c}=3000\mathrm{J}/\mathrm{kg}.\mathrm{K}$

The specific heat capacity of water is the amount of heat required to change the temperature of a unit mass of water by one degree.

We can use the concept of specific heat of the water and find the energy

transferred during this time. We can use the expression of the power consumed by the system and then use the expression of the heat of fusion and the specific heat of the sample in the frozen phase.

Formulae:

The heat energy required by a body, $\mathrm{Q}=\mathrm{mc}∆\mathrm{T}$ …(¾±)

Here, m is mass, c is specific heat capacity, $∆\mathrm{T}$is change in temperature, Q is required heat energy.

The heat energy released or absorbed by the body, $\mathrm{Q}={\mathrm{L}}_{\mathrm{F}}\mathrm{m}$ …(¾±¾±)

Here, L is specific latent heat of fusion.

The power exerted through the heat by a body, $\mathrm{P}=\frac{\mathrm{Q}}{\mathrm{t}}$ …(¾±¾±¾±)

Here, P is the power, Q is required heat energy, t is the time.

According to the figure, the temperature decreases from ${\mathrm{T}}_{\mathrm{i}}={300}^{°}\mathrm{C}$to ${\mathrm{T}}_{\mathrm{f}}={270}^{°}\mathrm{C}$within time, $\mathrm{t}=40.0\min$.

The energy transferred during this time using equation (i) is given by:

$\begin{array}{rcl}\mathrm{Q}& =& 3000\mathrm{J}/\mathrm{kg}.{}^{∘}\mathrm{C}×0.40\mathrm{kg}×\left|{270}^{∘}\mathrm{C}-{300}^{∘}\mathrm{C}\right|\\ & =& 36000\mathrm{J}\end{array}$

The transformation rate of the heat can be given using equation (ii) as:

$\begin{array}{rcl}\mathrm{P}& =& \frac{36000\mathrm{J}}{40\min }\\ & =& 900\frac{\mathrm{J}}{\min }\\ & =& \frac{900\mathrm{J}}{60\mathrm{s}}\\ & =& 15\mathrm{W}\end{array}$

During next $\mathrm{t}=30.0\min$, the heat transferred by the body using equation (iii) is given as:

$\begin{array}{rcl}\mathrm{Q}& =& 900\frac{\mathrm{J}}{\min }×30\min \\ & =& 27000\mathrm{J}\end{array}$

The phase changes from liquid to the solid. Then, the heat of transformation involved in this phase change is called heat of fusion. The specific latent heat of fusion using equation (ii) can be given as:

$\begin{array}{rcl}{\mathrm{L}}_{\mathrm{F}}& =& \frac{\mathrm{Q}}{\mathrm{m}}\\ & =& \frac{27000\mathrm{J}}{0.40\mathrm{kg}}\\ & =& 67500\mathrm{J}/\mathrm{kg}\\ & =& 68\mathrm{kJ}/\mathrm{kg}\end{array}$

Hence, the value of the specific heat of fusion is $68\mathrm{kJ}/\mathrm{kg}$.

During next $\mathrm{t}=20.0\min$, the sample is solid and change of temperature is $∆\mathrm{T}={30}^{∘}\mathrm{C}$. The expression for the specific heat of the sample using equation (i) is given as:

$\begin{array}{rcl}\mathrm{c}& =& \frac{\mathrm{Q}}{\mathrm{m}\left|∆\mathrm{T}\right|}\\ & =& \frac{27000\mathrm{J}}{0.40\mathrm{kg}×{30}^{∘}\mathrm{C}}\\ & =& 2250\frac{\mathrm{J}}{\mathrm{kg}.{}^{∘}\mathrm{C}}\\ & ≈& 2.3\frac{\mathrm{kJ}}{\mathrm{kg}.{}^{∘}\mathrm{C}}\end{array}$

Hence, the value of the specific heat is $2.3\frac{\mathrm{kJ}}{\mathrm{kg}.{}^{∘}\mathrm{C}}$