Answer to Question #162977 in Physics for Viv

Question

Archie drops 200 g of ice into 2.0 kg of water at 20 degrees * C . He stirs the water until all the ice has melted. He measures the temperature of the 2.2 kg of water to be 11 degrees * C (Specific heat capacity of water 4200 J/kg °C)

Answer 1

There is no question in this problem, so let us find the initial temperature of ice and see whether it is enough for 200 g of ice to cool the water to 11°C.

Heat received by ice (heating to 0°C, melting, and heating from 0 to 11°С):

$$Q_1=c_im_i(0-t_i)+m_i\lambda+c_wm_i(t-0).$$

Heat transferred by water to ice:

$$Q_2=c_wm_w(t-t_w).$$

In thermal equilibrium,

$$Q_1+Q_2=0,\\ c_im_i(0-t_i)+m_i\lambda+c_wm_i(t-0)+\\+c_wm_w(t-t_w)=0,\\ -c_im_it_i+m_i\lambda+c_wm_it+c_wm_w(t-t_w)=0,\\\space\\ t_i=\frac{m_i\lambda+c_w[m_it+m_w(t-t_w)]}{c_im_i}=1°\text{С}.$$

So, since ice cannot have the initial temperature of +1°C, 200 g of ice taken at 0°C would cool the water to a slightly lower temperature.