Pressure of an ideal gas, contained in a closed vessel, is increased by $0.4 \%$ when heated by $1^{\circ} \mathrm{C}$. Its initial temperature must be:

<p>In this scenario, we're dealing with an isochoric process, meaning the volume of the gas remains constant. For an ideal gas in an isochoric process, the pressure ($P$) is directly proportional to the temperature ($T$) in Kelvin.</p> <p>The relationship can be expressed as:</p> <p>$ P \propto T $</p> <p>Given the percentage increase in pressure is $0.4\%$ when the temperature is increased by $1^{\circ} \text{C}$, we can use the relationship:</p> <p>$ \frac{\Delta P}{P} = \frac{\Delta T}{T} $</p> <p>Substituting the given values into the equation, we have:</p> <p>$ \frac{0.4}{100} = \frac{1}{T} $</p> <p>Solving for $T$, we find:</p> <p>$ T = 250 \, \text{K} $</p> <p>Therefore, the initial temperature of the gas in the vessel must be $250 \, \text{K}$.</p>