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A cylinder with a movable piston is fill...

A cylinder with a movable piston is filled at `25^(@)C` with a gas that occupies a volume of `30.5 cm^(3)`. If the maximum capacity of the cylinder is `45.8 cm^(3)`, what is the highest temperature to which the cylinder can be heated at constant pressure without having the piston come out?

A

`50^(@)C`

B

`147.5^(@)C`

C

`174.5^(@)C`

D

`120^(@)C`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we will use Charles's Law, which states that for a given mass of gas at constant pressure, the volume of the gas is directly proportional to its absolute temperature (in Kelvin). The formula can be expressed as: \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \] Where: - \( V_1 \) = initial volume of the gas - \( T_1 \) = initial temperature of the gas in Kelvin - \( V_2 \) = final volume of the gas (maximum capacity of the cylinder) - \( T_2 \) = final temperature of the gas in Kelvin (which we want to find) ### Step 1: Convert the initial temperature to Kelvin The initial temperature \( T_1 \) is given as \( 25^\circ C \). To convert this to Kelvin, we use the formula: \[ T(K) = T(°C) + 273.15 \] So, \[ T_1 = 25 + 273.15 = 298.15 \, K \approx 298 \, K \] ### Step 2: Identify the volumes The initial volume \( V_1 \) is given as \( 30.5 \, cm^3 \) and the maximum volume \( V_2 \) is \( 45.8 \, cm^3 \). ### Step 3: Set up the equation using Charles's Law Using Charles's Law, we can set up the equation: \[ \frac{30.5}{298} = \frac{45.8}{T_2} \] ### Step 4: Solve for \( T_2 \) Cross-multiplying gives us: \[ 30.5 \cdot T_2 = 45.8 \cdot 298 \] Now, we can solve for \( T_2 \): \[ T_2 = \frac{45.8 \cdot 298}{30.5} \] Calculating the right side: \[ T_2 = \frac{13653.4}{30.5} \approx 447.4 \, K \] ### Step 5: Convert \( T_2 \) back to Celsius To convert \( T_2 \) back to Celsius, we use the formula: \[ T(°C) = T(K) - 273.15 \] So, \[ T_2(°C) = 447.4 - 273.15 \approx 174.25 \, °C \approx 174.4 \, °C \] ### Final Answer The highest temperature to which the cylinder can be heated at constant pressure without having the piston come out is approximately **174.4 °C**. ---
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