The ratio of molar heat capacities for an ideal gas is given as C_p...

Given the ratio of molar heat capacities $\gamma = \dfrac{C_p}{C_V} = \dfrac{7}{6}$

The molar heat capacity at constant volume $C_V$ is given by:

$C_V = \dfrac{R}{\gamma - 1} = \dfrac{R}{\dfrac{7}{6} - 1} = 6R$

The change in internal energy for an ideal gas depends only on the change in temperature and is given by:

$\Delta U = n C_V \Delta T$

Substituting the given values $n = 1.0 \text{ mol}$ and $\Delta T = 50 \text{ K}$:

$\Delta U = 1.0 \times 6R \times 50 = 300R$

Using the universal gas constant $R = 8.3 \text{ J mol}^{-1} \text{ K}^{-1}$:

$\Delta U = 300 \times 8.3 = 2490 \text{ J}$