$$\Delta V = V_f - V_i$$

where:

\( \Delta V \) is the change in volume

\(V_f\) is the final volume

\(V_i \) is the initial volume

We know that the initial temperature is \( \ T_i = 250.0 \ K \), and the initial volume is \( V_i = 1.95 L \). The final temperature is \( T_f = 442.2 K\).

We can use Charles's Law, which states that the volume of a gas is directly proportional to its temperature, provided that the pressure and number of moles remain constant:

$$V_f = V_i \frac{T_f}{T_i}$$

Substituting the given values, we get:

$$V_f = (1.95 L) x \frac{442.2 K}{250.0 K}$$

$$V_f = 3.54 L$$

Therefore, the change in volume is:

$$ \Delta V = V_f - V_i = 3.54 L - 1.95 L = 1.59 L$$

The volume of the nitrogen gas sample increases by 1.59 L when it is heated from 250.0 K to 442.2 K.