Question

When a gas filled in a closed vessel is heated by raising the temperature by 1ºC, its pressure increases by 0.4%. The initial temperature of the gas is _____ K.

Answer 1

Correct Answer: 250

To solve this problem, we need to determine the initial temperature of the gas in Kelvin when the pressure increases by 0.4% upon a 1ºC temperature increase. According to the ideal gas law for a constant volume and quantity of gas, we have: \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \). Given: \( \frac{P_2 - P_1}{P_1} = 0.004 \) and \( T_2 = T_1 + 1 \). Rearrange to find initial temperature:

Start with the relative change in pressure: \( P_2 = 1.004 \cdot P_1 \).
Substitute into the ideal gas law: \( \frac{P_1}{T_1} = \frac{1.004 \cdot P_1}{T_1 + 1} \).
Simplify: \( T_1 + 1 = 1.004 \cdot T_1 \).
Rearrange: \( 1 = 1.004 \cdot T_1 - T_1 \).
Simplify: \( 1 = 0.004 \cdot T_1 \).
Solve for \( T_1 \): \( T_1 = \frac{1}{0.004} = 250 \) K.

The calculated initial temperature is \( 250 \) K, which perfectly matches the expected range of 250,250. Therefore, the solution is verified.