Question

For water \(\Delta_{vap}H=41\) kJ mol\(^{-1}\) at 373 K and 1 bar pressure. Assuming that water vapour is an ideal gas that occupies a much larger volume than liquid water, the internal energy change during evaporation of water is __________ kJ mol\(^{-1}\). (Nearest integer)
[Use: R=8.3 J mol\(^{-1}\)K\(^{-1}\)]

Hint:

When using the formula \(\Delta H = \Delta U + \Delta n_g RT\), pay close attention to units. \(\Delta H\) is often given in kJ, while R is in J. You must convert them to the same unit before adding or subtracting. A common mistake is forgetting to convert kJ to J (or vice versa), leading to a significantly wrong answer.

Answer 1

Correct Answer: 38

Step 1: Understanding the Question
We are given the enthalpy of vaporization (\(\Delta_{vap}H\)) for water at its boiling point and asked to calculate the change in internal energy (\(\Delta U\)) for the same process.
Step 2: Key Formula or Approach
The relationship between enthalpy change (\(\Delta H\)) and internal energy change (\(\Delta U\)) is given by:
\[ \Delta H = \Delta U + P\Delta V \] For processes involving gases, this can be written as:
\[ \Delta H = \Delta U + \Delta n_g RT \] where \(\Delta n_g\) is the change in the number of moles of gas in the reaction.
Step 3: Detailed Calculation
Write the process and find \(\Delta n_g\):
The process is the evaporation of water:
\[ \text{H}_2\text{O}(l) \rightarrow \text{H}_2\text{O}(g) \] The change in the number of moles of gas is:
\[ \Delta n_g = (\text{moles of gaseous products}) - (\text{moles of gaseous reactants}) = 1 - 0 = 1 \] Rearrange the formula to solve for \(\Delta U\):
\[ \Delta U = \Delta H - \Delta n_g RT \] Substitute the given values and calculate \(\Delta U\):
\(\Delta H = 41 \text{ kJ mol}^{-1} = 41000 \text{ J mol}^{-1}\)
\(\Delta n_g = 1\)
\(R = 8.3 \text{ J mol}^{-1}\text{K}^{-1}\)
\(T = 373 \text{ K}\)
\[ \Delta n_g RT = (1 \text{ mol}) \times (8.3 \text{ J mol}^{-1}\text{K}^{-1}) \times (373 \text{ K}) = 3095.9 \text{ J} \] Now, calculate \(\Delta U\):
\[ \Delta U = 41000 \text{ J} - 3095.9 \text{ J} = 37904.1 \text{ J} \] Convert the answer to kJ and round to the nearest integer:
\[ \Delta U = \frac{37904.1}{1000} \text{ kJ} = 37.9041 \text{ kJ} \] Rounding to the nearest integer, we get 38 kJ. Step 4: Final Answer
The internal energy change is 38 kJ mol\(^{-1}\).