Calculate the final temperature. Two kg ice at zerodegree celsius is mixed with 8 kg of water at 20 degree celsius.
The heat of fusion of ice is 79.8 cal/deg C.
Let T = final temperature, and sum heat change = 0
8 kg (T-20) + 2 kg ((T-0)+(79.8))=0 cal-kg
Solve for T to get T=+0.04 deg. C.
To calculate the final temperature, we can use the principle of energy conservation, specifically the principle of heat exchange. The heat lost by the water at 20 degrees Celsius will be equal to the heat gained by the ice at 0 degrees Celsius until both reach a common final temperature.
We can use the formula:
\(Q_{\text{lost}} = Q_{\text{gained}}\)
Where:
\(Q_{\text{lost}}\) = heat lost by the water
\(Q_{\text{gained}}\) = heat gained by the ice
The equation for heat gained or lost is given by:
\(Q = mc\Delta T\)
Where:
\(Q\) = heat gained or lost
\(m\) = mass of the substance
\(c\) = specific heat capacity of the substance
\(\Delta T\) = change in temperature
For the water:
\(Q_{\text{lost}} = m_w c_w \Delta T_w\)
Where:
\(m_w\) = mass of water (8 kg)
\(c_w\) = specific heat capacity of water (4.186 J/g°C, or equivalently, 4186 J/kg°C)
\(\Delta T_w\) = change in temperature of water (\(T_{\text{final}} - 20\))
For the ice:
\(Q_{\text{gained}} = m_i c_i \Delta T_i\)
Where:
\(m_i\) = mass of ice (2 kg)
\(c_i\) = specific heat capacity of ice (2.093 J/g°C, or equivalently, 2093 J/kg°C)
\(\Delta T_i\) = change in temperature of ice (\(T_{\text{final}} - 0\))
Since the final temperature (T_final) is the same for both water and ice, we can rewrite the equations:
\(Q_{\text{lost}} = Q_{\text{gained}}\) \Rightarrow
\(m_w c_w (T_{\text{final}} - 20) = m_i c_i (T_{\text{final}} - 0)\)
Now, we can substitute the given values and solve for \(T_{\text{final}}\):
\(8 \times 4186 \times (T_{\text{final}} - 20) = 2 \times 2093 \times (T_{\text{final}} - 0)\)
Simplifying the equation:
\(33488(T_{\text{final}} - 20) = 4186(T_{\text{final}} - 0)\)
\(33488T_{\text{final}} - 667760 = 4186T_{\text{final}}\)
\(33488T_{\text{final}} - 4186T_{\text{final}} = 667760\)
\(29298T_{\text{final}} = 667760\)
\(T_{\text{final}} = \frac{667760}{29298}\)
\(T_{\text{final}} \approx 22.8\) degrees Celsius
Therefore, the final temperature of the mixture is approximately 22.8 degrees Celsius.