An ice cube having a mass of 50 g and an initial temperature of -20⁰C is place in 400 g of water 30⁰C water. What is the final temperature of the mixture if the effects of the container can be neglected?
the sum of the heats gained equals zero.
50(0-(-20)) +50*Lf +50(tf-0)+400(Tf-30)=0
you know the heat of fusion for ice (Lf), solve for Tf.
To find the final temperature of the mixture, we can use the principle of conservation of energy. The heat lost by the hot water is equal to the heat gained by the ice cube.
The heat lost by the hot water can be calculated using the equation:
Q_hot_water = m_hot_water * c_water * ΔT
Where:
m_hot_water = mass of hot water = 400 g
c_water = specific heat capacity of water = 4.18 J/g°C (approximately)
ΔT = change in temperature = final temperature - initial temperature
The heat gained by the ice cube can be calculated using the equation:
Q_ice = m_ice * c_ice * ΔT
Where:
m_ice = mass of ice = 50 g
c_ice = specific heat capacity of ice = 2.09 J/g°C (approximately)
ΔT = change in temperature = final temperature - initial temperature
Since the heat lost and gained are equal, we can equate the two equations and solve for the final temperature:
m_hot_water * c_water * ΔT = m_ice * c_ice * ΔT
Dividing both sides by ΔT:
m_hot_water * c_water = m_ice * c_ice
Substituting the given values:
400 g * 4.18 J/g°C = 50 g * 2.09 J/g°C
1669.6 J/°C = 104.5 J/°C
Now, we can solve for the final temperature:
1669.6 J/°C = 104.5 J/°C
ΔT = 1669.6 J/°C / 104.5 J/°C
ΔT ≈ 15.96 °C
To find the final temperature, we add the change in temperature to the initial temperature of the water:
Final temperature = Initial temperature + ΔT
Final temperature = 30°C + 15.96°C
Final temperature ≈ 45.96°C
Therefore, the final temperature of the mixture is approximately 45.96°C.
To find the final temperature of the mixture, we can use the principle of conservation of energy. The energy gained by the water must equal the energy lost by the ice cube.
First, let's calculate the heat gained by the water:
q_water = m_water * c_water * ΔT_water
where:
m_water = mass of water = 400 g
c_water = specific heat capacity of water = 4.18 J/g°C (this value is approximate)
ΔT_water = change in temperature of water = final temperature - initial temperature = final temperature - 30°C
Next, let's calculate the heat lost by the ice cube:
q_ice = m_ice * c_ice * ΔT_ice
where:
m_ice = mass of ice = 50 g
c_ice = specific heat capacity of ice = 2.09 J/g°C (this value is approximate)
ΔT_ice = change in temperature of ice = final temperature - initial temperature = final temperature - (-20°C)
Since the energy gained by the water is equal to the energy lost by the ice cube, we can set up the equation:
q_water = -q_ice
m_water * c_water * ΔT_water = -m_ice * c_ice * ΔT_ice
Substituting the values we know, we have:
400 * 4.18 * (final temperature - 30) = -50 * 2.09 * (final temperature - (-20))
Now we can solve for the final temperature.
1607.2 * (final temperature - 30) = -209 * (final temperature + 20)
1607.2 * final temperature - 48216 = -209 * final temperature - 4180
1607.2 * final temperature + 209 * final temperature = 48216 - 4180
1816.2 * final temperature = 44036
final temperature = 44036 / 1816.2
final temperature ≈ 24.2°C
Therefore, the final temperature of the mixture is approximately 24.2°C.