Suppose 0.0210 kg of steam (at 100.00°C) is added to 0.210 kg of water (initially at 19.5°C.). The water is inside a copper cup of mass 48.9 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.
use the following values:
c_copper=0.386 kJ/kg.K
c_water=4.19 kJ/kg.K
L_fusion=2.26 MJ/kg
To find the final temperature of the water after equilibrium is reached, we'll need to consider the heat transfer between the steam, water, and copper cup.
First, let's calculate the heat transferred from the steam to the water. We can use the formula:
Q = m * c * ΔT
Where:
Q = Heat transferred
m = mass
c = specific heat capacity
ΔT = change in temperature
For the steam, the initial temperature is 100.00°C, and the final temperature is the same as the final temperature of the water, which we're trying to find. The mass of the steam is 0.0210 kg, and the specific heat capacity of steam is approximately the same as that of water, which is 4.19 kJ/kg.K.
Let's calculate the heat transferred from the steam to the water:
Q_steam = 0.0210 kg * 4.19 kJ/kg.K * (final temperature - 100.0°C)
Next, let's calculate the heat transferred from the water to the copper cup. The change in temperature is the same for both the water and the copper cup, and we can use the formula:
Q = m * c * ΔT
The mass of the water is 0.210 kg, the specific heat capacity of water is 4.19 kJ/kg.K, and the initial temperature is 19.5°C. The copper cup has a mass of 48.9 g, which is equivalent to 0.0489 kg, and the specific heat capacity of copper is 0.386 kJ/kg.K.
Let's calculate the heat transferred from the water to the copper cup:
Q_water = (0.210 kg + 0.0489 kg) * 4.19 kJ/kg.K * (final temperature - 19.5°C)
Since the system is perfectly insulated, the heat transferred from the steam to the water must be equal to the heat transferred from the water to the copper cup. So we can equate these two quantities:
Q_steam = Q_water
0.0210 kg * 4.19 kJ/kg.K * (final temperature - 100.0°C) = (0.210 kg + 0.0489 kg) * 4.19 kJ/kg.K * (final temperature - 19.5°C)
Simplifying the equation:
0.0210 * (final temperature - 100.0) = (0.210 + 0.0489) * (final temperature - 19.5)
0.0210 * final temperature - 2.1 = 0.2589 * final temperature - 5.046255
0.0210 * final temperature - 0.2589 * final temperature = -2.1 -(-5.046255)
-0.2379 * final temperature = -2.1 + 5.046255
Now we can solve for the final temperature:
final temperature = (-2.1 + 5.046255) / (-0.2379)
final temperature ≈ 27.26°C
Therefore, the final temperature of the water after equilibrium is reached is approximately 27.26°C.