Suppose 0.0150 kg of steam (at 100.00°C) is added to 0.150 kg of water (initially at 19.3°C.). The water is inside a copper cup of mass 54.1 g. The cup is inside a perfectly insulated calorimetry container that prevents heat flow with the outside environment. Find the final temperature (in °C) of the water after equilibrium is reached.
The sum of the heats gained is zero.
Heatcodensingsteam+heatcondensedsteamTofinalTemp+heatwater to finaltemp+heatcopper to finaltemp=0
Heat gained condensing steam=-.00150*Hv the sign is negative because heat is LOST.
Heat gained by condensed steam=.0150(cw)(Tf-100)
Heat gained by heated water=.150*Cw*(Tf-19.3)
Heat gained by copper=.054(Ccu)(Tf-19.3)
add all these, set to zero, and solve for Tf.
To find the final temperature of the water after equilibrium is reached, we can apply the principle of conservation of energy.
First, let's consider the energy gained or lost by each component of the system:
1. The steam loses energy as it cools down from 100.00°C to the final temperature.
2. The water gains energy as it warms up from 19.3°C to the final temperature.
3. The copper cup gains energy as it warms up from the initial temperature to the final temperature.
The energy lost by the steam is given by:
Q_steam = mass_steam * specific_heat_steam * (final_temperature - initial_temperature_steam)
The energy gained by the water is given by:
Q_water = mass_water * specific_heat_water * (final_temperature - initial_temperature_water)
The energy gained by the copper cup is given by:
Q_cup = mass_cup * specific_heat_cup * (final_temperature - initial_temperature_cup)
Since the calorimetry container is perfectly insulated, there is no heat flow with the outside environment. Therefore, the sum of the energy gained and lost by each component of the system must be zero:
Q_steam + Q_water + Q_cup = 0
Plugging in the known values:
mass_steam = 0.0150 kg
initial_temperature_steam = 100.00°C
specific_heat_steam = specific heat capacity of steam (look up in a reference table)
mass_water = 0.150 kg
initial_temperature_water = 19.3°C
specific_heat_water = specific heat capacity of water (look up in a reference table)
mass_cup = 54.1 g (convert to kg by dividing by 1000)
initial_temperature_cup = initial temperature of water = 19.3°C
specific_heat_cup = specific heat capacity of copper (look up in a reference table)
final_temperature = unknown
Rearranging the equation and solving for the final temperature:
(final_temperature - initial_temperature_steam) * (mass_steam * specific_heat_steam) + (final_temperature - initial_temperature_water) * (mass_water * specific_heat_water) + (final_temperature - initial_temperature_cup) * (mass_cup * specific_heat_cup) = 0
Once you have the equation, you can plug in the known values and use algebraic methods to solve for the final temperature.
Note: The specific heat capacities of steam, water, and copper may vary with temperature, so if a temperature range is specified, you may need to use an average or interpolated value for the specific heat capacities.