A 75-g piece of aluminum at 89 °C is dropped in 1.0 L of water at 1.3 101 °C, which is in an insulated beaker. Assuming that there is negligible heat loss to the surroundings, determine the equilibrium temperature of the system
To determine the equilibrium temperature of the system, we need to use the principle of conservation of energy, specifically the conservation of heat.
First, let's determine the heat gained by the water. We can use the specific heat capacity formula:
q_water = m_water × c_water × ΔT_water
Where:
q_water is the heat gained by the water
m_water is the mass of water
c_water is the specific heat capacity of water (4.18 J/g·°C)
ΔT_water is the change in temperature of the water
Given:
- Volume of water (V_water) = 1.0 L
- Density of water = 1 g/mL (approximation)
- Mass of water (m_water) = V_water × density of water = 1.0 kg
ΔT_water can be calculated as the final temperature of the water (T_fwater) minus the initial temperature of the water (T_iwater):
ΔT_water = T_fwater - T_iwater
Now let's determine the heat lost by the aluminum. We can use the same formula:
q_aluminum = m_aluminum × c_aluminum × ΔT_aluminum
Where:
q_aluminum is the heat lost by the aluminum
m_aluminum is the mass of aluminum (75 g)
c_aluminum is the specific heat capacity of aluminum (0.897 J/g·°C)
ΔT_aluminum is the change in temperature of the aluminum
ΔT_aluminum can be calculated as the final temperature of the system (T_fsystem) minus the initial temperature of the aluminum (T_ialuminum):
ΔT_aluminum = T_fsystem - T_ialuminum
Since the system reaches equilibrium, the heat gained by the water is equal to the heat lost by the aluminum:
q_water = q_aluminum
m_water × c_water × ΔT_water = m_aluminum × c_aluminum × ΔT_aluminum
Substituting the known values:
(1.0 kg) × (4.18 J/g·°C) × ΔT_water = (75 g) × (0.897 J/g·°C) × ΔT_aluminum
Now let's solve for ΔT_water:
ΔT_water = (75 g × 0.897 J/g·°C × ΔT_aluminum) / (1.0 kg × 4.18 J/g·°C)
Simplifying:
ΔT_water = (67.275 J·°C × ΔT_aluminum) / 4.18 J·°C
Next, we can use this value of ΔT_water to find the equilibrium temperature of the system:
T_fsystem = T_iwater + ΔT_water
Given:
T_iwater = 101 °C
Finally, we can substitute the values to calculate the equilibrium temperature of the system.