If you combine 34 kg of ice at 0°C with 80 kg of steam at 100°C, what is the final temperature of the system?
°C
nice solution
To determine the final temperature of the system after combining the ice and steam, we can use the principle of energy conservation.
First, let's calculate the energy required to heat the ice to its melting point (0°C):
Q1 = (mass of ice) × (specific heat capacity of ice) × (change in temperature)
= 34 kg × 2.09 J/g°C × (0°C - (-0°C))
= 34 kg × 2.09 J/g°C × 0°C
= 0 J
Since the temperature change is zero, no energy is required to heat the ice.
Next, let's calculate the energy required to melt the ice at its melting point (0°C):
Q2 = (mass of ice) × (heat of fusion of ice)
= 34 kg × 334 J/g
= 34 kg × 334,000 J/kg
= 11,356,000 J
Now, let's calculate the energy required to heat the water from its melting point (0°C) to the final temperature (T):
Q3 = (mass of water) × (specific heat capacity of water) × (change in temperature)
= 34 kg × 4.18 J/g°C × (T - 0°C)
= 1412.12 J/g°C × T
Similarly, let's calculate the energy required to heat the steam to its final temperature (T):
Q4 = (mass of steam) × (specific heat capacity of steam) × (change in temperature)
= 80 kg × 2.03 J/g°C × (T - 100°C)
= 162.4 J/g°C × (T - 100°C)
Now, applying the principle of energy conservation:
Q1 + Q2 + Q3 = Q4
0 J + 11,356,000 J + 1412.12 J/g°C × T = 162.4 J/g°C × (T - 100°C)
Simplifying the equation:
11,356,000 J + 1412.12 J/g°C × T = 162.4 J/g°C × T - 16,240 J
Solving for T:
1412.12 J/g°C × T - 162.4 J/g°C × T = 11,356,000 J - 16,240 J
1249.72 J/g°C × T = 11,339,760 J
T = 11,339,760 J / 1249.72 J/g°C
T ≈ 9077.3 °C
Therefore, the final temperature of the system is approximately 9077.3 °C.