Question

If one mole of ammonia and one mole of hydrogen chloride is mixed in a closed container to form ammonium chloride gas, then:
[A]$\Delta H>\Delta U$
[B]$\Delta H=\Delta U$
[C]$\Delta H<\Delta U$
[D] There is no relationship

Answer 1

Hint: We can solve this by using the equation which gives us the relation between change in internal energy and enthalpy. From the ideal gas law equation we can replace the pressure and volume and get the equation in terms of number of moles and temperature. By putting the number of moles of reactant and product, we can find the change in the number of moles and use that to assume whether $\Delta H$is positive, negative or equal to change in internal energy.

Complete step by step answer:
$\Delta $H is the enthalpy. It is the amount of heat released or evolved in a reaction at a constant pressure.
A negative value of enthalpy means heat is released in the reaction and positive value of enthalpy means energy was required to carry out the reaction.
It is related to internal energy and change in volume as-
     \[\Delta H=\Delta U+P\Delta V\]
Where, $\Delta U$ is the change in internal energy of the system and
$\Delta V$is the change in volume,
P is the pressure.
For an exothermic reaction, change in enthalpy is negative and for an endothermic reaction, the change in enthalpy is positive.
As we can see, the reaction given to us in the question is occurring in a closed container. When a reaction takes place in a closed container, it means neither the product nor the reactant can escape from the system i.e. the reaction is taking place in a closed system.
Therefore, in a closed system the volume will be fixed.
As we know, PV=nRT, which is the ideal gas law.
We can also solve this by replacing PV by nRT in the equation.
We can write that, \[\Delta H=\Delta U + \Delta nRT\]where,
\[\Delta n\] is the change in the number of moles,
R is the universal gas constant and T is the temperature.
We can write the given reaction as-
     \[N{{H}_{3}}(g)+HCl(g)\to N{{H}_{4}}Cl(g)\]
\[\Delta n\] = number of moles of product – number of moles of reactant = 1 - 2 = -1.
Therefore, \[\Delta H=\Delta U+(-1)RT = \Delta U-RT\]
As we know R has a fixed positive value for every gas and temperature also has a fixed value. Therefore, a positive quantity is subtracted from the change in internal energy to get the change in enthalpy which means $\Delta $H is less than$\Delta U$.
Therefore, the correct answer is option [C]$\Delta H<\Delta U$.

Note: In a closed system, there is no exchange of matter with the surrounding but there can be change in energy as heat or work. Whereas in an open system there can be exchange of matter as well as energy. This is the first law of thermodynamics.