Question
Give the relationship between $\Delta U$ and $\Delta H$ for gases.
✓
Answer
Let $V _{ A }$ be the total volume of gaseous reactants,
$V _{ B }$ be the total volume of gaseous product.
Let $n _{ A }$ be the number of moles of the reactant,
$n _{ B }$ be the number of moles of the product,
At constant pressure and temperature,
$
\begin{aligned}
& p V_A=n_A R T, \\
& p V_B=n_B R T \\
& \Rightarrow pV_{B}-pV_{A}=\left(n_{B}-n_{A}\right) RT \\
& \Rightarrow p \Delta V=(\Delta n)_g RT
\end{aligned}
$
Here, $(\Delta n)_g=n_B-n_A$ is equal to the difference between the number of moles of gaseous products and gaseous reactants.
We know that,
$
\Delta H=\Delta U+(\Delta n)_g R T
$
Now, $\Delta H = q _{ p }$ (heat change under constant pressure),
$\Delta U = q _{ v }$ (heat change under constant volume).
Therefore, $q_p=q_v+(\Delta n)_g R T$
$V _{ B }$ be the total volume of gaseous product.
Let $n _{ A }$ be the number of moles of the reactant,
$n _{ B }$ be the number of moles of the product,
At constant pressure and temperature,
$
\begin{aligned}
& p V_A=n_A R T, \\
& p V_B=n_B R T \\
& \Rightarrow pV_{B}-pV_{A}=\left(n_{B}-n_{A}\right) RT \\
& \Rightarrow p \Delta V=(\Delta n)_g RT
\end{aligned}
$
Here, $(\Delta n)_g=n_B-n_A$ is equal to the difference between the number of moles of gaseous products and gaseous reactants.
We know that,
$
\Delta H=\Delta U+(\Delta n)_g R T
$
Now, $\Delta H = q _{ p }$ (heat change under constant pressure),
$\Delta U = q _{ v }$ (heat change under constant volume).
Therefore, $q_p=q_v+(\Delta n)_g R T$
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