A 2.75kg block of copper at 22C dropped into 5.25kg of water at 12.0C. What is the final temperature of the mixture?
Specific heat:
water = 4.19J/kgK
copper = 390J/kgK
To find the final temperature of the mixture, we can use the principle of conservation of energy.
First, let's calculate the heat gained by the copper block:
Q_copper = mass_copper * specific_heat_copper * change_in_temperature_copper
where:
mass_copper = 2.75 kg (mass of the copper block)
specific_heat_copper = 390 J/kgK (specific heat of copper)
change_in_temperature_copper = (final temperature - initial temperature of copper)
Next, let's calculate the heat gained by the water:
Q_water = mass_water * specific_heat_water * change_in_temperature_water
where:
mass_water = 5.25 kg (mass of the water)
specific_heat_water = 4.19 J/kgK (specific heat of water)
change_in_temperature_water = (final temperature - initial temperature of water)
According to the principle of conservation of energy, the heat gained by the copper block is equal to the heat lost by the water:
Q_copper = -Q_water
Substituting the above equations, we get:
mass_copper * specific_heat_copper * change_in_temperature_copper = -mass_water * specific_heat_water * change_in_temperature_water
Rearranging the equation, we find:
change_in_temperature_water / change_in_temperature_copper = - (mass_copper * specific_heat_copper) / (mass_water * specific_heat_water)
Now we can plug in the given values:
change_in_temperature_water / change_in_temperature_copper = - (2.75 kg * 390 J/kgK) / (5.25 kg * 4.19 J/kgK)
change_in_temperature_water / change_in_temperature_copper = - (1072.5 J/K) / (21.9775 J/K)
change_in_temperature_water / change_in_temperature_copper ≈ - 48.73
Since the negative sign indicates that the water loses heat and the copper gains heat, the ratio of temperature changes should be positive. We can ignore the negative sign for now.
Therefore:
change_in_temperature_water / change_in_temperature_copper ≈ 48.73
To find the final temperature of the mixture, we need to know the ratio of temperature changes. Assuming the final temperature of the mixture is denoted as T_final, we can write:
T_final - 12°C / T_final - 22°C = 48.73
Solving this equation will give us the value of T_final.
Please note that this calculation assumes no heat loss to the surroundings and that the specific heat capacities remain constant over the given temperature range.
To find the final temperature of the mixture, we need to use the principle of heat transfer and the equation for heat gained or lost.
The equation for heat transfer is:
Q = mcΔT
Where:
Q is the heat gained or lost
m is the mass of the substance
c is the specific heat capacity of the substance
ΔT is the change in temperature
In this case, the copper block is losing heat and the water is gaining heat. We can calculate the heat lost by the copper block and the heat gained by the water separately, and then set them equal to each other to find the final temperature.
Step 1: Calculate the heat lost by the copper block:
Q_copper = mcΔT
m_copper = 2.75 kg (mass of copper block)
c_copper = 390 J/kgK (specific heat of copper)
ΔT_copper = (final temperature - initial temperature of copper block)
Step 2: Calculate the heat gained by the water:
Q_water = mwΔT
m_water = 5.25 kg (mass of water)
c_water = 4.19 J/kgK (specific heat of water)
ΔT_water = (final temperature - initial temperature of water)
Step 3: Set the heat lost and heat gained equal to each other:
Q_copper = Q_water
mcΔT = mwΔT
Step 4: Solve for the final temperature:
mcΔT = mwΔT
(2.75 kg)(390 J/kgK)(final temperature - initial temperature of copper block) = (5.25 kg)(4.19 J/kgK)(final temperature - initial temperature of water)
Simplify and solve for the final temperature.
Since the initial temperature of the copper block and water are given (22°C and 12°C) respectively, you can substitute these values into the equation and solve for the final temperature.