# Consider the reaction ${{N}_{2}}(g)+3{{H}_{2}}(g)->2N{{H}_{3}}(g)$carried out at constant pressure and temperature. If \[\mathbf{\Delta }H\]and \[\mathbf{\Delta }U\]are enthalpy change and internal energy change respectively, then which of the following expressions is true?

A.\[\mathbf{\Delta }H=0\]

B.\[\mathbf{\Delta }H=\mathbf{\Delta }U\]

C.\[\mathbf{\Delta }H<\mathbf{\Delta }U\]

D.\[\mathbf{\Delta }H>\mathbf{\Delta }U\]

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**Hint:**Try to find out the difference in the number of moles of product and reactants. Then try to put it in an equation which relates $\Delta H,\Delta U$ and number of moles, n. An analysis can be drawn afterwards, which will lead to the answer.

**Complete answer:**

Enthalpy is an energy-like property or state function, which means that it has the dimensions of energy (and is thus measured in units of joules or ergs), and its value is set entirely by the temperature, pressure, and composition of the system and not by its history. In symbols, the enthalpy, H, equals the sum of the inner energy, E, and also the product of the pressure, P, and volume, V, of the system. According to the law of energy conservation, the change in internal energy is up to the warmth transferred to, less the work done by, the system. If the sole work done may be a change of volume at constant pressure, the enthalpy change is strictly adequate the warmth transferred to the system. When energy must be added to a fabric to alter its phase from a liquid to a gas, that quantity of energy is termed the enthalpy (or latent heat) of vaporization and is expressed in units of joules per mole. Other phase transitions have similar associated enthalpy changes, like the enthalpy (or latent heat) of fusion for changes from a solid to a liquid. like other energy functions, it's neither convenient nor necessary to work out absolute values of enthalpy. Now, we have the reaction:

${{N}_{2}}+3{{H}_{2}}->2N{{H}_{3}}$ (all are in gaseous form)

Let $\Delta n$be the difference of moles between products and reactants.

So we obtain $\Delta n$= 2-(3+1)= -2

Now, by using the equation,

\[\mathbf{\Delta }H-\mathbf{\Delta }U+\mathbf{\Delta }nRT\], we have ( R=universal gas constant , T= temperature)

\[\mathbf{\Delta }H=\mathbf{\Delta }U-2RT\], as $\Delta n$=2

Now let us observe the equation. The values of R are constant and T is also constant as it is given in the question. That means, the term ‘2RT’ is a positive term. When a positive term is getting subtracted from a bigger positive number i.e $\Delta U$ we obtain$\Delta H$, which is also positive. So, $\Delta H<\Delta U$is the required condition which gives option C as the answer.

**Note:**

Here, the temperature is assumed to be positive as generally equations are written in SI units. The SI unit of temperature is kelvin and the least value of kelvin is 0, which is also a positive number.