When a gas filled in a closed vessel is heated by raising the temperature by 1°C, its pressure increases by 0.4%. What is the initial temperature of the gas in K?
Answer (1)
To find the initial temperature of the gas, we'll use the concept of the ideal gas law, which relates pressure, volume, and temperature. For a gas in a closed vessel with no change in volume, the relationship can be expressed with the formula for isochoric (constant volume) processes: T 1 P 1 = T 2 P 2 .
Given:
The temperature is increased by 1\, ^{\circ}\mathrm{C} .
The pressure increases by 0.4% .
Solution:
Express the initial and final conditions :
Let the initial pressure be P 1 and the initial temperature be T 1 (in Kelvin).
After heating, the new temperature T 2 = T 1 + 1 .
The new pressure would be P 2 = P 1 + 0.004 P 1 = 1.004 P 1 .
Apply the isochoric process equation :
T 1 P 1 = T 1 + 1 1.004 P 1
Since P 1 cancels out from the equation, we have:
T 1 1 = T 1 + 1 1.004
Solve for T 1 : Cross-multiply: T 1 + 1 = 1.004 T 1
Simplify this:
T 1 + 1 = 1.004 T 1
\[1 = 1.004T_1 - T_1\]
\[1 = 0.004T_1\]
\[T_1 = \frac{1}{0.004} = 250\]
Therefore, the initial temperature of the gas is 250 K .
