**Finding the Initial Temperature of the Gas**

To find the initial temperature of the gas in the closed vessel, we can use the ideal gas law equation: PV = nRT, where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature.

**Using the Ideal Gas Law Equation**

Let's assume that the volume and the number of moles of the gas remain constant. Therefore, we can rewrite the ideal gas law equation as:

P1/T1 = P2/T2

Where P1 is the initial pressure, T1 is the initial temperature, P2 is the final pressure, and T2 is the final temperature.

**Applying the Given Information**

From the given information, we know that the pressure of the gas increases by 0.4% when heated by 1 degree Celsius. We can represent this as:

P2 = P1 + 0.004P1
ΔT = 1°C

Substituting these values into the equation, we have:

(P1 + 0.004P1)/T2 = P1/T1

**Simplifying the Equation**

Next, we can simplify the equation by canceling out the common factor P1:

(1 + 0.004)/T2 = 1/T1

Simplifying further:

1.004/T2 = 1/T1

**Solving for the Initial Temperature**

To find T1, we can cross multiply and solve for T1:

T1 = T2 / (1.004)

Since T2 is the final temperature after heating, we can find the initial temperature by subtracting the change in temperature from T2:

T1 = (T2 - ΔT) / (1.004)

Therefore, the initial temperature of the gas in the closed vessel can be found by subtracting the change in temperature (1 degree Celsius) from the final temperature (T2) and dividing it by 1.004.

**Summary**

To find the initial temperature of the gas in a closed vessel, we can use the ideal gas law equation. By applying the given information of a 0.4% increase in pressure when heated by 1 degree Celsius, we can derive an equation and solve for the initial temperature. The equation allows us to find the initial temperature by subtracting the change in temperature from the final temperature and dividing it by 1.004.