Question

When a gas-filled closed vessel is heated to $1°\mathrm{C}$, its pressure increases by $0.4\%$. What is the initial temperature of the gas?

Answer 1

Step 1: Given data

Temperature is $1^{°}\mathrm{C}$

Increased percentage is $0.4\%$

Here the volume is constant.

Step 2: Formula used

Gay-Lussac's Law:

1. The pressure exerted by a gas (with a given mass and preserved at a fixed volume) varies directly with the absolute temperature of the gas, according to Gay-Lussac's Law.
2. In other words, while the mass is constant and the volume is constant, the pressure produced by a gas is proportional to the temperature of the gas.

Mathematically, Gay-Lussac's law can be represented as:

$P\propto T$

Here,

$\mathrm{P}$ is the pressure.

$\mathrm{T}$ is the temperature.

Step 3: Calculating the initial temperature of the gas,

$$\begin{gathered} \Rightarrow \frac{{\mathrm{P}}_1}{{\mathrm{P}}_2}=\frac{{\mathrm{T}}_1}{{\mathrm{T}}_2} \\ \Rightarrow \frac{\mathrm{P}}{\mathrm{P}+0.004\mathrm{P}}=\frac{\mathrm{T}+273}{\mathrm{T}+274} \\ \Rightarrow \frac{P}{1.004\mathrm{P}}=\frac{\mathrm{T}+273}{\mathrm{T}+274} \\ \Rightarrow \mathrm{T}=-23°\mathrm{C} \\ =250\mathrm{K} \end{gathered}$$

The initial temperature of the gas is $250\mathrm{K}$.