[[First law of thermodynamics]]
# First law for isobaric processes
$$
\begin{align*}
dW &= -p\, dV \\
dQ &= n C_p\, dT
\end{align*}
$$
where

- $C_p : \pu{J.mol-1.K-1}$ is the **molar specific heat for gas of constant pressure**

## For ideal gas
By the [[Ideal gas law]], if $p$ is constant,
$$
\begin{align*}
dV = \frac{nR}{p}\, dT
\end{align*}
$$
and therefore
$$
\begin{align*}
dW &= - nR\, dT \\
dU &= dQ + dW \\
&= n C_p\, dT - R\, dT
\end{align*}
$$
which, using the equation from [[First law for isovolumetric processes]],
shows
$$
\begin{align*}
nC_V\, dT &= n C_p\, dT - R\, dT \\
C_V &= C_p - R
\end{align*}
$$
### Monatomic
As a consequence of the above,
for a monatomic ideal gas
$$
\begin{align*}
C_p = \frac{5}{2} R
\end{align*}
$$
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