Question

The enthalpy of vaporization of water at \[{{100}^{\circ }}C\] is $40.63\,KJ\text{ }mo{{l}^{-1}}$ The value $\Delta U$ for this process:
(a)- $37.35\text{ }KJ\text{ }mo{{l}^{-1}}$
(b)- $39.08KJ\text{ }mo{{l}^{-1}}$
(c)- $42.19KJ\text{ }mo{{l}^{-1}}$
(d)- $43.73KJ\text{ }mo{{l}^{-1}}$

Answer 1

Hint: The internal energy of a system is calculated by the formula $\Delta H=\Delta U+\Delta {{n}_{g}}RT$ where $\Delta H$ is the change in enthalpy of the system, $\Delta U$ is the change in internal energy of the system, R is the gas constant, T is the temperature of the system, and $\Delta {{n}_{g}}$ is a difference of the number of moles of product and reactants.

Complete step by step answer:
Internal energy change is the heat absorbed or evolved at constant volume. It is denoted by $\Delta U$.
Enthalpy change of a system is equal to the heat absorbed or evolved by the system at constant pressure. It is denoted by $\Delta H$.

The internal energy change and change in enthalpy can be related as $\Delta H=\Delta U+\Delta {{n}_{g}}RT$
Where, $\Delta H$ is the change in enthalpy of the system and the question it is given $40.63\, KJ\text{ }mo{{l}^{-1}}$
R is the gas constant and its value is taken $8.314\text{ x 1}{{\text{0}}^{-3}}\text{ J}{{\text{K}}^{-1}}mo{{l}^{-1}}$
T is the temperature of the system.
It is always taken in Kelvin (K).

The temperature of the water is given \[{{100}^{\circ }}C\]. It has to be converted into Kelvin.
$T=100+273=373K$
The temperature is 373 K.

$\Delta {{n}_{g}}$ is the difference between the number of moles of products and reactants.
So, the reaction is:
${{H}_{2}}O(l)\to {{H}_{2}}O(g)$
$\Delta {{n}_{g}}={{n}_{p}}-{{n}_{r}}$

So, the number of moles in the product is 1, and the number of moles in the reactant is 0 because the moles of liquid and solid are taken 0.
$\Delta {{n}_{g}}=1-0=1$
So, the internal energy change can be calculated.
$\Delta H=\Delta U+\Delta {{n}_{g}}RT$
$\Delta U=\Delta H-\Delta {{n}_{g}}RT$
$\Delta U=40.63-(1\ \text{x 8}\text{.314 x 1}{{\text{0}}^{-3}}\text{ x 373)}$
$\Delta U=37.53 KJ/mol$
So, the correct answer is “Option A”.

Note: The number of moles must be taken correctly. Only the moles of gaseous elements or compounds are taken for the calculation. The temperature must be converted to Kelvin if present in other forms.