Question

In an insulated container, 0.50 kg of water at 80°C is mixed with 0.050 kg of ice at −5.0°C. After a while, all the ice melts, leaving only the water. Find the final temperature $T_f$ of the water. The freezing point of water is 0°C.

Answer 1

Explanation:

When ice tends to absorb heat energy then it will change into liquid state. Now,  change in the temperature of cold water $dT = T_{f} - 0^{o}C$ = $T_{f}$

Change in the temperature of original water in the container $dT' = 80 -T_{f }$

Latent heat of melting of ice L = 80 cal/g

Specific heat of water C = 1 cal/g C

Heat absorbed by ice to melt = mass($m_{i}$) x latent heat(L)

Heat absorbed by water = mass($m_{w}$) x specific heat(C) x change in temperature (dT')

$m_{i}$ = 0.050 kg = 50 g and $m_{w}$ = 0.50 kg = 500 g

Now, heat lost by water is as follows.

heat lost by hot water( at $80^{o}C$)  = heat absorbed by the ice + heat absorbed by cold water

         $m_{i} \times L + m_{i} \times C dT = m_{w} \times C dT'$

            $50 \times 80 + 50 \times 1 \times (T_{f}) = 500 \times 1 \times (80 - T_{f})$

So, we will cancel out 50 from each term given above. Hence,

              $80 + T_{f} = 800 - 10T_{f}$  

               $11 T_{f}$ = 720

               $T_{f} = 65.45^{o}C$

Thus, we can conclude that final temperature of the water is $65.45^{o}C$.

Answer 2

Answer:

$T_f=68.808\ ^{\circ}C$

Explanation:

Given:

• mass of water, $m_w=0.5\ kg$

• initial temperature of water, $T_{wi}=80^{\circ}C$

• mass of ice, $m_i=0.05\ kg$

• initial temperature of ice, $T_{ii}=-5^{\circ}C$

• Specific heat capacity of water, $c_w=4186\ J.kg^{-1}.K^{-1}$

• Specific heat capacity of ice, $c_i=2100\ J.kg^{-1}.K^{-1}$

• Latent heat of fusion of ice, $L=335000\ J.kg^{-1}$

Now according to the given that the whole ice melts:

using energy conservation with heat,

$m_i.c_i.(T_f-T_i)+m_i.L+m_w.c_w.(T_f-T_i)=0$

$m_i.c_i.(T_f-T_i)+m_i.L=m_w.c_w.(T_i-T_f)$

$0.05\times 2100\times (T_f-(-5))+0.05\times 335000=0.5\times 4186\times (80-T_f)$

$T_f=68.808\ ^{\circ}C$