Step 1: Understanding the Concept:
The gas is contained in a closed vessel, which means its volume remains constant ($V = \text{constant}$).
According to Gay-Lussac's law, for a fixed mass of an ideal gas at constant volume, the pressure is directly proportional to its absolute temperature ($P \propto T$).
Step 2: Key Formula or Approach:
From $P \propto T$, we can write:
\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]
For small percentage changes, we can also use the fractional change relation:
\[ \frac{\Delta P}{P} = \frac{\Delta T}{T} \]
Step 3: Detailed Explanation:
Let the initial pressure be $P$ and the initial temperature be $T$.
The pressure is increased by $2.5%$.
The fractional change in pressure is:
\[ \frac{\Delta P}{P} = \frac{2.5}{100} = 0.025 \]
The temperature increases by $4 \text{ K}$, so the change in temperature is:
\[ \Delta T = 4 \text{ K} \]
Using the fractional change formula (which is exact here because it is a direct linear proportionality):
\[ \frac{\Delta P}{P} = \frac{\Delta T}{T} \]
Substitute the known values:
\[ 0.025 = \frac{4}{T} \]
Rearrange the equation to solve for $T$:
\[ T = \frac{4}{0.025} \]
Multiply numerator and denominator by 1000 to remove the decimal:
\[ T = \frac{4000}{25} \]
\[ T = 160 \text{ K} \]
Alternatively, using the exact state equations:
$P_2 = P + 0.025P = 1.025P$
$T_2 = T + 4$
$\frac{P}{T} = \frac{1.025P}{T + 4} \implies 1.025T = T + 4 \implies 0.025T = 4 \implies T = 160 \text{ K}$.
Step 4: Final Answer:
The initial temperature of the gas is 160 K.