practice activity # 3\nsolve the following problem. show complete details of your answers.\nhow much heat is…
practice activity # 3\nsolve the following problem. show complete details of your answers.\nhow much heat is required to change 500g of water at 100 °c into vapor at 108 °c?
Answer
To solve the problem of finding the heat required to change 500g of water at (100^\circ \text{C}) into vapor at (108^\circ \text{C}), we need to consider two processes:
- Phase change (liquid water to water vapor at (100^\circ \text{C}))
- Sensible heating (heating the water vapor from (100^\circ \text{C}) to (108^\circ \text{C}))
Step 1: Phase Change (Latent Heat of Vaporization)
The latent heat of vaporization of water, (L_v), is the heat required to convert liquid water to vapor at its boiling point ((100^\circ \text{C})) without changing temperature. For water, (L_v = 2.26 \times 10^6 , \text{J/kg}) (or (2260 , \text{J/g})).
The mass of water, (m = 500 , \text{g} = 0.5 , \text{kg}).
The heat required for phase change, (Q_1), is given by:
[
Q_1 = m \cdot L_v
]
Substitute (m = 0.5 , \text{kg}) and (L_v = 2.26 \times 10^6 , \text{J/kg}):
[
Q_1 = 0.5 , \text{kg} \times 2.26 \times 10^6 , \text{J/kg} = 1.13 \times 10^6 , \text{J}
]
Step 2: Sensible Heating of Vapor
After vaporization, we heat the water vapor from (100^\circ \text{C}) to (108^\circ \text{C}). The specific heat capacity of water vapor, (c_v), is approximately (2000 , \text{J/(kg·°C)}) (or (2 , \text{J/(g·°C)})).
The temperature change, (\Delta T = 108^\circ \text{C} - 100^\circ \text{C} = 8^\circ \text{C}).
The heat required for sensible heating, (Q_2), is given by:
[
Q_2 = m \cdot c_v \cdot \Delta T
]
Substitute (m = 0.5 , \text{kg}), (c_v = 2000 , \text{J/(kg·°C)}), and (\Delta T = 8^\circ \text{C}):
[
Q_2 = 0.5 , \text{kg} \times 2000 , \text{J/(kg·°C)} \times 8^\circ \text{C} = 8000 , \text{J}
]
Step 3: Total Heat Required
The total heat, (Q_{\text{total}}), is the sum of (Q_1) and (Q_2):
[
Q_{\text{total}} = Q_1 + Q_2
]
Substitute (Q_1 = 1.13 \times 10^6 , \text{J}) and (Q_2 = 8000 , \text{J}):
[
Q_{\text{total}} = 1.13 \times 10^6 , \text{J} + 8000 , \text{J} = 1.138 \times 10^6 , \text{J}
]
Answer:
The total heat required is (\boldsymbol{1.138 \times 10^6 , \text{J}}) (or (1138000 , \text{J})).