. Determine the temperature, T that results when 150 g of ice at 0°C is mixed with 300 g of water at 50°C. Given heat of fusion, l of water = 3.35 × 105 J/kg, cwater = 4186 J/kg∙oC.
Sum of heats gained is zero (some gains, some loses).
Heat ice + heat water=0
150*Hfice+150*cw*(Tf-0)+300*cw*(Tf-50)=0
solve for Tfinal, Tf
To determine the final temperature, we need to calculate the heat gained and heat lost in the system.
First, let's calculate the heat gained by the ice:
q_ice = mass_ice * heat_of_fusion
= 150 g * 3.35 × 10^5 J/kg
= 5.025 × 10^7 J
Next, let's calculate the heat gained by the water:
q_water = mass_water * specific_heat_capacity_water * ΔT_water
= 300 g * 4186 J/kg∙oC * (T_final - 50°C)
Now, let's equate the heat gained by the ice to the heat gained by the water and solve for the final temperature (T_final):
5.025 × 10^7 J = 300 g * 4186 J/kg∙oC * (T_final - 50°C)
Let's solve this equation step by step:
1. Simplify the equation:
5.025 × 10^7 J = 1255800 J/°C * (T_final - 50°C)
2. Divide both sides of the equation by 1255800 J/°C:
(T_final - 50°C) = (5.025 × 10^7 J) / (1255800 J/°C)
≈ 40°C
3. Solve for T_final:
T_final = 40°C + 50°C
= 90°C
Therefore, the final temperature of the mixture is 90°C.
To determine the final temperature when ice and water are mixed, we need to calculate the energy gained or lost by each substance and then equate it. This can be done using the principle of conservation of energy.
We'll follow these steps to solve the problem:
Step 1: Calculate the heat gained or lost during the phase change of the ice (from solid to liquid).
Step 2: Calculate the heat gained or lost during the temperature change of the ice (from 0°C to T).
Step 3: Calculate the heat gained or lost during the temperature change of the water (from 50°C to T).
Step 4: Set up an equation equating the total heat gained and lost for both substances.
Step 5: Solve the equation for the final temperature, T.
Let's go through each step in detail:
Step 1: Calculate the heat gained or lost during the phase change of the ice (from solid to liquid):
The heat of fusion (Lf) is the amount of energy required to change 1 kg of a substance from a solid to a liquid at its melting point. In this case, we are given Lf for water, which is 3.35 × 10^5 J/kg.
The ice is at 0°C and will need to be heated to 0°C to melt. Therefore, the heat gained or lost during the phase change of the ice can be calculated using the formula:
Q_ice = m_ice * L_f
where Q_ice is the heat gained or lost during the phase change of the ice, and m_ice is the mass of the ice.
Substituting the given values, we have:
Q_ice = (150 g / 1000) * (3.35 × 10^5 J/kg)
Step 2: Calculate the heat gained or lost during the temperature change of the ice (from 0°C to T):
To calculate the heat gained or lost during the temperature change of the ice, we can use the heat capacity equation:
Q_ice_temp = m_ice * c_ice * (T - 0)
where Q_ice_temp is the heat gained or lost during the temperature change of the ice, c_ice is the specific heat capacity of ice (we'll assume it's the same as the specific heat capacity of water, which is 4186 J/kg∙°C), and T is the final temperature.
Substituting the given values and the calculated Q_ice from step 1, we have:
Q_ice_temp = (150 g / 1000) * 4186 J/kg∙°C * (T - 0)
Step 3: Calculate the heat gained or lost during the temperature change of the water (from 50°C to T):
To calculate the heat gained or lost during the temperature change of the water, we can use the heat capacity equation:
Q_water_temp = m_water * c_water * (T - 50)
where Q_water_temp is the heat gained or lost during the temperature change of the water, m_water is the mass of the water, and c_water is the specific heat capacity of water (which is given as 4186 J/kg∙°C).
Substituting the given values, we have:
Q_water_temp = (300 g / 1000) * 4186 J/kg∙°C * (T - 50)
Step 4: Set up an equation equating the total heat gained and lost for both substances:
According to the principle of conservation of energy, the total heat gained by the ice and water is equal to the total heat lost by the ice and water, so we can set up an equation:
Q_ice + Q_ice_temp = Q_water_temp
Substituting the calculated values from steps 1, 2, and 3, we have:
(150 g / 1000) * (3.35 × 10^5 J/kg) + (150 g / 1000) * 4186 J/kg∙°C * (T - 0) = (300 g / 1000) * 4186 J/kg∙°C * (T - 50)
Step 5: Solve the equation for the final temperature, T:
Now you can solve the equation for T. Simplify and solve the equation using algebraic manipulation to isolate T on one side of the equation. Once you have solved for T, you will have the final temperature when the ice and water are mixed.