A copper calorimeter can with mass .100 kg contains .160 kg of water and .018 kg of ice in thermal equilibrium at atmospheric pressure. If .750 kg of lead at a temperature of 255 degrees Celsius is dropped into the can what is the final temperature of the system if no heat is lost to the surroundings?
The sum of all the heats gained is zero (some of the heats gained will be negative, as in heat lost).
Heatgainedwater+ heatgainedlead+ heatgained calorimeter+ heat gained ice=0
mw*cw*(Tf-0) + ml*cl*(Tf-255) + mc*cc*(tf-0) + massice*latentheatfusionice =0
Solve for Tf
To find the final temperature of the system, we need to use the principle of conservation of energy, which states that the energy lost by one object is equal to the energy gained by another object.
First, let's calculate the energy lost by the lead when it cools down to the final temperature, using the specific heat capacity of lead (c_lead):
Energy lost by lead = mass_lead * specific heat capacity_lead * change in temperature_lead
where:
mass_lead = 0.750 kg (mass of the lead)
specific heat capacity_lead = 128 J/(kg·°C) (specific heat capacity of lead)
change in temperature_lead = (final temperature - initial temperature)
Next, let's calculate the energy gained by the water and ice when they warm up to the final temperature, using the specific heat capacity of water (c_water) and the specific latent heat of fusion (L_fusion) of ice:
Energy gained by water = mass_water * specific heat capacity_water * change in temperature_water
Energy gained by ice (to melt) = mass_ice * specific latent heat of fusion
where:
mass_water = 0.160 kg (mass of water)
specific heat capacity_water = 4186 J/(kg·°C) (specific heat capacity of water)
change in temperature_water = (final temperature - initial temperature)
mass_ice = 0.018 kg (mass of ice)
specific latent heat of fusion = 334,000 J/kg (specific latent heat of fusion of ice)
Now, since no heat is lost to the surroundings, the energy lost by the lead must be equal to the sum of the energy gained by the water and ice in order to reach thermal equilibrium:
Energy lost by lead = Energy gained by water + Energy gained by ice
Using this principle, we can equate the equations above and solve for the final temperature.
Final Step:
Solve the equation for the final temperature. Rearrange the terms and isolate the final temperature on one side of the equation. Once you have the equation in this form, plug in the known values and calculate the final temperature.
It's important to note that this calculation assumes all heat transfer is purely conductive, and there are no other heat losses or gains within the system.