Question

When is $\mathbf{\Delta H}\mathbf{=}\frac{\mathbf{5}}{\mathbf{2}}\mathbf{RT}$? When is $\mathbf{\Delta E}\mathbf{=}\frac{\mathbf{5}}{\mathbf{2}}\mathbf{RT}$? When is $\mathbf{\Delta H}\mathbf{=}\frac{\mathbf{3}}{\mathbf{2}}\mathbf{RT}$? When is $\mathbf{\Delta H}\mathbf{=}\mathbf{\Delta E}$?What does this say, if anything, about$\mathbf{△}\mathbf{E}$and$\mathbf{△}\mathbf{H}$as state functions?

Short answer

$$\begin{gathered} \mathrm{\Delta H}={\mathrm{C}}_{\mathrm{p}}\mathrm{\Delta T} \\ \mathrm{\Delta H}=\frac{5}{2}\mathrm{R\Delta T} \\ \mathrm{\Delta E}={\mathrm{C}}_{\mathrm{p}}\mathrm{\Delta T} \\ \mathrm{\Delta E}=\frac{3}{2}\mathrm{R\Delta T} \end{gathered}$$

The fact that $\mathrm{\Delta E}$and $\mathrm{\Delta H}$can be calculated using only the initial and final temperature shows that $\mathrm{\Delta E}$and $\mathrm{\Delta H}$will only depend on the initial and final states of the system and not how the system moves between states, which is expected for state functions.

Step-by-step solution

Enthalpy

The sum of internal energy and the product of pressure and volume is referred to as enthalpy. It is a state function as the change in enthalpy does not depend on the pathway between two states.

Explanation

For one mole of a monoatomic ideal gas, and can be calculated using the following formulas as:

$$\begin{gathered} \mathrm{\Delta H}={\mathrm{C}}_{\mathrm{p}}\mathrm{\Delta T} \\ \mathrm{\Delta H}=\frac{5}{2}\mathrm{R\Delta T} \\ \mathrm{\Delta E}={\mathrm{C}}_{\mathrm{p}}\mathrm{\Delta T} \\ \mathrm{\Delta E}=\frac{3}{2}\mathrm{R\Delta T} \end{gathered}$$

The process where the temperature does not change, an isothermal process will have$\mathrm{\Delta H}=\mathrm{\Delta E}$because both the values will have equal zero as $\mathrm{\Delta T}=0$.

The fact that $\mathrm{\Delta H}\mathrm{and}\mathrm{\Delta E}$and can be calculated using only the initial final temperature shows that $\mathrm{\Delta H}\mathrm{and}\mathrm{\Delta E}$and will only depend on the initial and final states of the system and not how the system move between states, which is expected for state functions.