A person makes iced tea by adding ice to 1.8 kg of hot tea, initially at $80 ^ { \circ } \mathrm { C }$ How many kilograms of ice, initially at $0 ^ { \circ } \mathrm { C }$ are required to bring the mixture to $10 ^ { \circ } \mathrm { C } ?$ The specific heat of water (and tea) is $4186 \mathrm {~J} / \mathrm { kg } \cdot \mathrm { K }$ and the latent heat of fusion of ice is $3.34 \times 10 ^ { 5 } \mathrm {~J} / \mathrm { kg }$

Question

A person makes iced tea by adding ice to  1.8 kg of hot tea, initially at $80 ^ { \circ } \mathrm { C }$ How many kilograms of ice, initially at $0 ^ { \circ } \mathrm { C }$ are required to bring the mixture to $10 ^ { \circ } \mathrm { C } ?$ The specific heat of water (and tea) is $4186 \mathrm {~J} / \mathrm { kg } \cdot \mathrm { K }$ and the latent heat of fusion of ice is $3.34 \times 10 ^ { 5 } \mathrm {~J} / \mathrm { kg }$

Quizplus · Accepted Answer

To find the mass of ice needed, we use the principle of conservation of energy. The heat lost by the tea as it cools down must equal the heat gained by the ice as it melts and then warms up to the final temperature. The heat lost by the tea is given by $Q_{\text{lost}} = m_{\text{tea}} \cdot c_{\text{water}} \cdot \Delta T_{\text{tea}}$, where $m_{\text{tea}} = 1.8 \, \text{kg}$, $c_{\text{water}} = 4186 \, \text{J/kg} \cdot \text{K}$, and $\Delta T_{\text{tea}} = 80 ^{\circ}\text{C} - 10 ^{\circ}\text{C} = 70 \, \text{K}$. The heat gained by the ice is the sum of the heat required to melt the ice ($Q_{\text{melt}} = m_{\text{ice}} \cdot L_{\text{fusion}}$) and the heat required to warm the resulting water to the final temperature ($Q_{\text{warm}} = m_{\text{ice}} \cdot c_{\text{water}} \cdot \Delta T_{\text{water}}$), where $L_{\text{fusion}} = 3.34 \times 10^5 \, \text{J/kg}$ and $\Delta T_{\text{water}} = 10 ^{\circ}\text{C} - 0 ^{\circ}\text{C} = 10 \, \text{K}$. Setting $Q_{\text{lost}} = Q_{\text{melt}} + Q_{\text{warm}}$ and solving for $m_{\text{ice}}$ yields the correct mass of ice required.

A Person Makes Iced Tea by Adding Ice to 80∘C80 ^ { \circ } \mathrm { C }80∘C

A Person Makes Iced Tea by Adding Ice to $80 ^ { \circ } \mathrm { C }$