To solve this problem, we can apply the principle of conservation of energy. The heat lost by the hot water will be equal to the heat gained by the ice and the resulting water at the final temperature. We'll use the specific heat capacity of water, the specific latent heat of fusion of ice, and the masses and initial temperatures of the water and ice.
Given: Mass of ice (m_ice) = 1 kg Mass of water (m_water) = 9 kg Initial temperature of water (T_water_initial) = 50°C Specific latent heat of fusion of ice (L_fusion) = 336,000 J/kg
Let's assume the final temperature of the water and ice mixture is T_final.
First, we'll calculate the heat gained by the ice as it melts: Q_ice = m_ice * L_fusion
Next, we'll calculate the heat lost by the hot water: Q_water = m_water * c_water * (T_water_initial - T_final)
where c_water is the specific heat capacity of water (approximately 4,186 J/kg°C).
Since energy is conserved, the heat gained by the ice will be equal to the heat lost by the water:
Q_ice = Q_water
Substituting the values and rearranging the equation, we have:
m_ice * L_fusion = m_water * c_water * (T_water_initial - T_final)
Now we can solve for T_final:
T_final = T_water_initial - (m_ice * L_fusion) / (m_water * c_water)
Plugging in the given values:
T_final = 50 - (1 * 336,000) / (9 * 4,186)
Calculating the value:
T_final ≈ 50 - 8.03 ≈ 41.97°C
Therefore, the final temperature of the water will be approximately 41.97°C after the ice has melted.