You cool a 100g slug of red hot iron (745oC) by dropping it into an insulated cup of negligible mass containing 75.0g of water, at 20oC. Assuming no heat exchange with the surroundings (a) what is the final temperature of the iron? (b) what is the final mass of the iron and the remaining water?
final mass of iron in water 167
To solve this problem, we will use the principle of conservation of energy. The energy lost by the iron will be gained by the water:
(a) To find the final temperature of the iron, we can use the equation:
mass_iron * specific_heat_iron * (final_temperature_iron - initial_temperature_iron) = mass_water * specific_heat_water * (final_temperature_water - initial_temperature_water)
Here,
mass_iron = 100g (mass of iron)
specific_heat_iron = 0.449 J/g°C (specific heat capacity of iron)
final_temperature_iron = ?
initial_temperature_iron = 745°C (red hot iron temperature)
mass_water = 75.0g (mass of water)
specific_heat_water = 4.184 J/g°C (specific heat capacity of water)
final_temperature_water = ?
Let's plug in the values and solve for the final temperature of the iron:
100g * 0.449 J/g°C * (final_temperature_iron - 745°C) = 75.0g * 4.184 J/g°C * (20°C - initial_temperature_water)
Simplifying the equation:
44.9 (final_temperature_iron - 745) = 313.8 (20 - initial_temperature_water)
Now, we need the initial temperature of the water. Since it is given as 20°C, we'll substitute that value:
44.9 (final_temperature_iron - 745) = 313.8 (20 - 20)
Thus, the equation simplifies to:
44.9 (final_temperature_iron - 745) = 0
final_temperature_iron - 745 = 0
final_temperature_iron = 745°C
Therefore, the final temperature of the iron is 745°C.
(b) To find the final mass of the iron and remaining water, we'll use the conservation of mass principle:
mass_iron + mass_water = final_mass_iron + final_mass_water
Let's substitute the known values:
100g + 75.0g = final_mass_iron + final_mass_water
175.0g = final_mass_iron + final_mass_water
Since the iron does not evaporate and no other change occurs, the final mass of the iron remains the same:
final_mass_iron = 100g
Now we can calculate the final mass of the water:
final_mass_water = 175.0g - final_mass_iron
final_mass_water = 175.0g - 100g
final_mass_water = 75.0g
Therefore, the final mass of the iron is 100g, and the remaining water is 75.0g.
To find the final temperature of the iron and the final mass of the iron and water, we can use the principle of conservation of energy and the specific heat capacity of the substances involved.
(a) Final temperature of the iron:
To find the final temperature of the iron, we need to calculate the energy exchanged between the iron and water.
First, we need to calculate the energy released by the cooling iron using the equation:
Q_iron = m_iron * c_iron * ΔT_iron
where:
Q_iron is the energy released by the iron
m_iron is the mass of the iron (100g)
c_iron is the specific heat capacity of iron (0.45 J/g⋅°C)
ΔT_iron is the change in temperature of the iron (final temperature - initial temperature)
The initial temperature of the iron is 745°C, and the final temperature is unknown. Let's denote the final temperature of the iron as T_final.
Q_iron = 100g * 0.45 J/g⋅°C * (T_final - 745°C)
The energy absorbed by the water can be calculated using the equation:
Q_water = m_water * c_water * ΔT_water
where:
Q_water is the energy absorbed by the water
m_water is the mass of the water (75.0g)
c_water is the specific heat capacity of water (4.18 J/g⋅°C)
ΔT_water is the change in temperature of the water (final temperature - initial temperature)
The initial temperature of the water is 20°C, and the final temperature is also T_final.
Q_water = 75.0g * 4.18 J/g⋅°C * (T_final - 20°C)
Since there is no heat exchange with the surroundings (insulated cup), the energy released by the iron is equal to the energy absorbed by the water:
Q_iron = Q_water
100g * 0.45 J/g⋅°C * (T_final - 745°C) = 75.0g * 4.18 J/g⋅°C * (T_final - 20°C)
Now we can solve this equation to find the final temperature of the iron (T_final).
(b) Final mass of the iron and remaining water:
To find the final mass of the iron and remaining water, we need to consider the conservation of mass. The mass of the iron and water before and after the cooling process should remain the same.
Initial mass of iron + mass of water = Final mass of iron + mass of remaining water
100g + 75.0g = Final mass of iron + mass of remaining water
To find the final mass of the iron and remaining water, we need to know how much iron is dissolved in the water. Without this information, we cannot give a definite answer to part (b) of the question.
Therefore, we can only calculate the final temperature of the iron using the information given.