Question

practice activity # 3
solve the following problem. show complete details of your answers.
how much heat is required to change 500g of water at 100 °c into vapor at 108 °c?

Explanation:

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:

1. Phase change (liquid water to water vapor at \(100^\circ \text{C}\))
2. 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 \(1138000 \, \text{J}\)).