A container contains equal masses of $\mathrm{H}_2, \mathrm{He}, \mathrm{CO}_2$ and Ne at a certain temperature. Which of the following gases exerts minimum partial pressure?

According to Dalton’s law, the partial pressure of a gas is proportional to its mole fraction, or equivalently, to its number of moles when volume and temperature are constant. Given equal masses $m$ of $\mathrm{H}_2$, $\mathrm{He}$, $\mathrm{CO}_2$, and $\mathrm{Ne}$, the number of moles for each is calculated as $n = \frac{m}{M}$, where $M$ is the molar mass.

The molar masses are $M_{\mathrm{H}_2} = 2 \, \text{g/mol}$, $M_{\mathrm{He}} = 4 \, \text{g/mol}$, $M_{\mathrm{CO}_2} = 44 \, \text{g/mol}$, and $M_{\mathrm{Ne}} = 20 \, \text{g/mol}$. Thus, the moles are $n_{\mathrm{H}_2} = \frac{m}{2}$, $n_{\mathrm{He}} = \frac{m}{4}$, $n_{\mathrm{CO}_2} = \frac{m}{44}$, and $n_{\mathrm{Ne}} = \frac{m}{20}$.

Since $m$ is constant, the number of moles is inversely proportional to molar mass. The largest molar mass belongs to $\mathrm{CO}_2$ ($44 \, \text{g/mol}$), so it has the smallest number of moles and therefore the minimum partial pressure.

Final answer: $\boxed{\text{C}}$