A piece of iron has a mass of 0.4 kg and is initially at 500 degrees. It is lowered into a beaker with 20L of water at 22 degrees. What is the final equilibrium temperature? Assume there is no heat loss to the environment during the process.
To find the final equilibrium temperature, we need to use the principle of heat transfer known as the heat equation:
Q = mcΔT
where Q is the heat transferred, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.
Let's break down the steps to solve this problem:
Step 1: Calculate the heat transfer from the iron to the water.
The heat transferred from the iron to the water can be calculated using the equation:
Q_iron = m_iron * c_iron * ΔT_iron
The specific heat capacity of iron (c_iron) is around 450 J/kg°C.
The change in temperature for the iron (ΔT_iron) can be calculated using the equation:
ΔT_iron = T_final - T_initial
T_initial = 500°C (given)
To find T_final, we need to calculate the heat transfer for the water first.
Step 2: Calculate the heat transfer from the water to the iron.
Similarly, the heat transferred from the water to the iron can be calculated using the equation:
Q_water = m_water * c_water * ΔT_water
The specific heat capacity of water (c_water) is approximately 4186 J/kg°C.
The change in temperature for the water (ΔT_water) can be calculated using the equation:
ΔT_water = T_initial - T_final
T_initial = 22°C (given)
Since there is no heat loss to the environment during the process, the total heat transferred from the iron to the water should be equal.
Step 3: Equate the heat transferred. Since both the heat transfer values are equal to each other, we can write:
Q_iron = Q_water
Step 4: Substitute the values into the equation.
Substituting the respective values into the equation, we have:
m_iron * c_iron * ΔT_iron = m_water * c_water * ΔT_water
Step 5: Solve for T_final.
Rearranging the equation to solve for T_final, we get:
T_final = (m_iron * c_iron * T_initial + m_water * c_water * T_initial) / (m_iron * c_iron + m_water * c_water)
Now, let's substitute the given values:
m_iron = 0.4 kg
T_initial = 500°C
m_water = 20 kg (20 liters of water is approximately equal to 20 kg)
T_initial = 22°C
c_iron = 450 J/kg°C
c_water = 4186 J/kg°C
Substituting these values into the equation, we can calculate the final equilibrium temperature.
To determine the final equilibrium temperature, we can use the principle of conservation of energy. The total heat gained by the water is equal to the total heat lost by the iron.
Let's break down the problem step-by-step:
1. Determine the heat lost by the iron:
- We can use the specific heat equation: Q = mcΔT, where Q is the heat transferred, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.
- The specific heat capacity of iron is approximately 450 J/kg°C.
- The initial temperature of the iron is 500°C, and the final temperature is unknown.
- The mass of the iron is given as 0.4 kg.
Therefore, the heat lost by the iron can be calculated using the formula:
Q_lost = m * c * (T_initial - T_final)
2. Determine the heat gained by the water:
- The specific heat capacity of water is 4186 J/kg°C.
- The initial temperature of the water is 22°C.
- The volume of water is given as 20 liters, which is equivalent to 20 kg since the density of water is approximately 1000 kg/m^3.
Therefore, the heat gained by the water can be calculated using the formula:
Q_gained = m * c * (T_final - T_initial)
Now, we equate the heat lost by the iron to the heat gained by the water:
Q_lost = Q_gained
m_iron * c_iron * (T_initial_iron - T_final) = m_water * c_water * (T_final - T_initial_water)
Substituting the given values:
0.4 kg * 450 J/kg°C * (500°C - T_final) = 20 kg * 4186 J/kg°C * (T_final - 22°C)
Simplifying the equation:
180 kg°C * (500°C - T_final) = 83720 kg°C * (T_final - 22°C)
Solving the equation:
90000 °C - 180 kg°C * T_final = 83720 kg°C * T_final - 1839840 kg°C
90000 °C + 1839840 kg°C = 83720 kg°C * T_final + 180 kg°C * T_final
1939840 kg°C = 101900 kg°C * T_final
T_final = 19°C
Therefore, the final equilibrium temperature is 19 degrees Celsius.