[[First law of thermodynamics]]
# First Law for isovolumetric processes
When the volume of a gas is constant,
no mechanical transfer of energy takes place
and therefore no work is done on or by the gas
(see [[Quasistatic expression for work]]).
Hence
$$
\begin{align*}
dQ &= dU \\
&= n C_V\, dT
\end{align*}
$$
where

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

## For monatomic ideal gas
Since $U = \frac{3}{2} n R T$
$$
\begin{align*}
dU = \frac{3}{2} n R \, dT = n C_V dT
\end{align*}
$$
hence
$$
\begin{align*}
C_V =& \frac{3}{2}  R \\
&\approx \pu{12.5 J.mol-1.K-1}
\end{align*}
$$

---
#state/develop