A mass M of solid metal at temperature T is put into 3.0kg of water
which is initially at 90◦C. The metal and water are placed in an insulated
container.
The specific heat for water is 4186 J
kg K . The metal has a melting point
of 60◦C. The latent heat associated with the solid-liquid transition of the
metal is 7.5 × 104 J
kg . The specific heat for the metal is 600 J
kg in the solid
phase and 400 J
kg in the liquid phase
Suppose that the initial temperature of the metal is T = −20◦C and
that M = 15kg. What is the final equilibrium temperature?
Suppose that the final equilibrium state occurred at a temperature of
60◦C (the melting point of the metal). Assuming that the metal was
initially solid at 0◦C, what range of M is consistent with this final
state?
Suppose that the final state is liquid metal in equilibrium with water
at 75◦C. Given that the mass of the metal is M = 2.0kg, what was
the initial temperature T?
I will be happy to review your work
To solve these problems, we will apply the principle of conservation of energy.
1. What is the final equilibrium temperature?
In this case, we need to calculate the heat gained by the water and the heat lost by the metal, and set them equal to each other.
Let's denote the final equilibrium temperature as T' and the heat gained/lost as Q.
The heat gained by the water is given by:
Q_water = m_water * c_water * (T' - 90)
The heat lost by the metal is given by:
Q_metal = m_metal * c_solid * (60 - T) + m_metal * L_fusion + m_metal * c_liquid * (T' - 60)
where c_solid is the specific heat of the metal in the solid phase (600 J/kg K), L_fusion is the latent heat associated with the solid-liquid transition of the metal (7.5 x 10^4 J/kg), and c_liquid is the specific heat of the metal in the liquid phase (400 J/kg K).
Setting the two equations equal to each other and solving for T', we have:
m_water * c_water * (T' - 90) = m_metal * c_solid * (60 - T) + m_metal * L_fusion + m_metal * c_liquid * (T' - 60)
Plugging in the values given in the problem, we have:
3.0 kg * 4186 J/kg K * (T' - 90) = 15 kg * 600 J/kg K * (60 - (-20)) + 15 kg * 7.5 x 10^4 J/kg + 15 kg * 400 J/kg K * (T' - 60)
Simplifying and solving for T':
12558(T' - 90) = 582000 + 1125000 + 6000(T' - 60)
12558T' - 1130220 = 582000 + 1125000 + 6000T' - 360000
11958T' = 1955220
T' ≈ 163.33°C
Therefore, the final equilibrium temperature is approximately 163.33°C.
2. What range of M is consistent with a final equilibrium state at 60°C?
For the final equilibrium state to be at 60°C, the metal must have completely melted. This means that the heat gained by the water is equal to the heat lost by the metal.
Let's denote the mass of the metal as M and solve for the range of M.
The heat gained by the water is given by:
Q_water = m_water * c_water * (60 - 90)
The heat lost by the metal is given by:
Q_metal = M * c_solid * (60 - 0) + M * L_fusion + M * c_liquid * (60 - 60)
Setting the two equations equal to each other and solving for M, we have:
3.0 kg * 4186 J/kg K * (-30) = M * 600 J/kg K * 60 + M * 7.5 x 10^4 J/kg + M * 400 J/kg K * 0
-376740 = 36000M + 75000M
-376740 = 111000M
M ≈ -3.39 kg
Since mass cannot be negative, there is no range of M that is consistent with a final equilibrium state at 60°C.
3. What was the initial temperature T if the final state is liquid metal in equilibrium with water at 75°C?
In this case, the heat gained by the water is equal to the heat lost by the metal.
Let's denote the initial temperature of the metal as T and solve for T.
The heat gained by the water is given by:
Q_water = m_water * c_water * (75 - 90)
The heat lost by the metal is given by:
Q_metal = M * c_solid * (60 - T) + M * L_fusion + M * c_liquid * (75 - 60)
Setting the two equations equal to each other and solving for T, we have:
3.0 kg * 4186 J/kg K * (-15) = 2.0 kg * 600 J/kg K * (60 - T) + 2.0 kg * 7.5 x 10^4 J/kg + 2.0 kg * 400 J/kg K * (75 - 60)
-188760 = 12000 - 1200T + 150000 + 800(T - 15)
-188760 = 12000 - 1200T + 150000 + 800T - 12000
-188760 = 800T - 1200T + 150000 - 12000
102240T = 260760
T ≈ 2.55°C
Therefore, the initial temperature T is approximately 2.55°C.
To solve these questions, we need to consider the heat transfer that occurs between the metal and the water. The heat transferred can be calculated using the equation:
Q = mcΔT
Where Q is the heat transferred, m is the mass of the substance, c is the specific heat capacity, and ΔT is the change in temperature.
Let's analyze each question separately:
1. What is the final equilibrium temperature?
To find the final equilibrium temperature, we need to calculate the total heat transferred between the metal and water and set it equal to zero (since equilibrium has been reached):
Q_hot + Q_cold = 0
The heat transferred from the hot object to the cold object is given by:
Q_hot = mc_hotΔT_hot
Since we know the specific heat capacities of the metal and water, we can calculate the heat transferred for each:
Q_hot = (M)(c_metal)(T_hot - T_eq)
The heat transferred from the cold object (water) is given by:
Q_cold = mc_coldΔT_cold
Q_cold = (m_water)(c_water)(T_cold - T_eq)
Setting these two equations equal to each other and solving for T_eq will give us the final equilibrium temperature.
2. What range of M is consistent with the final state at a temperature of 60◦C?
In this case, the final equilibrium state occurs at the melting point of the metal. We need to calculate the total heat transferred between the metal and water, assuming the metal was initially solid at 0◦C. The formula for heat transferred is the same as in question 1.
Once we calculate the total heat transferred, we can use the given latent heat associated with the solid-liquid transition to determine if the metal will completely melt at 60◦C. If it does, we can find the range of M that is consistent with this final state.
3. What was the initial temperature T?
Here, we are given the final state of liquid metal in equilibrium with water at 75◦C. We need to calculate the initial temperature T of the metal.
Like in the previous questions, we calculate the total heat transferred between the metal and water. However, this time we consider the final temperatures:
Q_hot = (M)(c_metal)(T_initial - T_final)
Q_cold = (m_water)(c_water)(T_final - T_initial)
Setting these two equations equal to each other and solving for T_initial will give us the initial temperature of the metal.
By following these steps and using the provided formulas and values, you can find the answers to the three questions.