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A gas is present in a cylinder fitted wi...

A gas is present in a cylinder fitted with movable piston. Above and below the piston there is equal number of moles of gas. The volume above is two times the volume below at a temperature of 300 K. At what temperature will the volume above be four times the volume below-

A

600 K

B

400 K

C

200 K

D

120 K

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the situation using the ideal gas law and the relationship between temperature and volume for a gas under constant pressure conditions. ### Step-by-Step Solution: 1. **Understanding the Initial Conditions**: - We have a cylinder with a movable piston. - The volume above the piston (V_upper) is 2V, and the volume below the piston (V_lower) is V. - The initial temperature (T_initial) is 300 K. - The number of moles of gas (n) is equal above and below the piston. 2. **Setting Up the Final Condition**: - We need to find the final temperature (T_final) when the volume above the piston becomes four times the volume below the piston. - Therefore, the new volume above the piston (V_upper_final) will be 4V, and the volume below the piston (V_lower_final) remains V. 3. **Applying the Ideal Gas Law**: - The ideal gas law states that \( PV = nRT \). - Since the pressure (P) is constant (due to the movable piston), we can relate the initial and final states using the formula: \[ \frac{T_{initial}}{V_{initial}} = \frac{T_{final}}{V_{final}} \] 4. **Substituting the Known Values**: - For the initial state: - \( T_{initial} = 300 \, K \) - \( V_{initial} = V_{upper} = 2V \) - For the final state: - \( T_{final} = ? \) - \( V_{final} = V_{upper\_final} = 4V \) 5. **Setting Up the Equation**: - Plugging in the values into the equation: \[ \frac{300}{2V} = \frac{T_{final}}{4V} \] 6. **Solving for T_final**: - Cancel out the common volume (V) from both sides: \[ \frac{300}{2} = \frac{T_{final}}{4} \] - This simplifies to: \[ 150 = \frac{T_{final}}{4} \] - Now, multiply both sides by 4 to find \( T_{final} \): \[ T_{final} = 150 \times 4 = 600 \, K \] ### Final Answer: The temperature at which the volume above the piston will be four times the volume below is **600 K**.
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Knowledge Check

  • If the volumes of the two boxes above are equal, what is the value of h?

    A
    `1`
    B
    `2`
    C
    `4`
    D
    `5`
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