1 g of water in liquid phase has volume `1 cm^(3)` and in vapour phase 1671 `cm^(3)` at atmospheric pressure and the latent heat of vaporization of water is 2256 J/g, the change in the internal energy in joules for 1 g of water of 373K when it changes from liquid phase to vapour phase at the same temperature is:

Options

A. 2256
B. 167
C. 2089
D. 1

Answer

To find the change in internal energy when 1 g of water changes from liquid to vapor at 373 K, we can use the first law of thermodynamics, which states:

\[ \Delta U = Q - W \]

where:
- \(\Delta U\) is the change in internal energy,
- \(Q\) is the heat added to the system,
- \(W\) is the work done by the system.

### Step 1: Calculate the Heat Added (Q)
The heat added when 1 g of water vaporizes can be calculated using the formula:

\[ Q = m \cdot L \]

where:
- \(m\) is the mass of water (1 g),
- \(L\) is the latent heat of vaporization (2256 J/g).

Substituting the values:

\[ Q = 1 \, \text{g} \times 2256 \, \text{J/g} = 2256 \, \text{J} \]

### Step 2: Calculate the Work Done (W)
The work done by the system when the water vaporizes can be calculated using the formula:

\[ W = P \cdot \Delta V \]

where:
- \(P\) is the atmospheric pressure (approximately \(10^5 \, \text{N/m}^2\)),
- \(\Delta V\) is the change in volume.

The initial volume of liquid water is \(1 \, \text{cm}^3\) and the final volume of vapor is \(1671 \, \text{cm}^3\). Thus, the change in volume is:

\[ \Delta V = V_{\text{final}} - V_{\text{initial}} = 1671 \, \text{cm}^3 - 1 \, \text{cm}^3 = 1670 \, \text{cm}^3 \]

Converting \(1670 \, \text{cm}^3\) to \(m^3\):

\[ 1670 \, \text{cm}^3 = 1670 \times 10^{-6} \, m^3 = 1.67 \times 10^{-3} \, m^3 \]

Now, substituting the values into the work done equation:

\[ W = 10^5 \, \text{N/m}^2 \times 1.67 \times 10^{-3} \, m^3 = 167 \, \text{J} \]

### Step 3: Calculate the Change in Internal Energy (ΔU)
Now, we can substitute the values of \(Q\) and \(W\) into the first law of thermodynamics equation:

\[ \Delta U = Q - W = 2256 \, \text{J} - 167 \, \text{J} = 2089 \, \text{J} \]

### Final Answer
The change in internal energy when 1 g of water changes from liquid to vapor at 373 K is:

\[ \Delta U = 2089 \, \text{J} \]