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?

Options

A. `50^(@)C`
B. `147.5^(@)C`
C. `174.5^(@)C`
D. `120^(@)C`

Answer

To solve the problem, we will use the relationship between volume and temperature for a gas at constant pressure, which is given by Charles's Law. The formula we will use is:

\[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]

Where:
- \( V_1 \) = initial volume of the gas
- \( T_1 \) = initial temperature of the gas (in Kelvin)
- \( V_2 \) = maximum volume of the gas
- \( T_2 \) = final temperature of the gas (in Kelvin)

### Step 1: Convert the initial temperature from Celsius to Kelvin
To convert Celsius to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273.15 \]
Given \( T_1 = 25°C \):
\[ T_1 = 25 + 273.15 = 298.15 \, K \]

### Step 2: Identify the volumes
From the problem statement:
- \( V_1 = 30.5 \, cm^3 \)
- \( V_2 = 45.8 \, cm^3 \)

### Step 3: Rearrange the Charles's Law equation to solve for \( T_2 \)
We can rearrange the equation to find \( T_2 \):
\[ T_2 = \frac{V_2 \cdot T_1}{V_1} \]

### Step 4: Substitute the known values into the equation
Now we can substitute \( V_2 \), \( T_1 \), and \( V_1 \) into the equation:
\[ T_2 = \frac{45.8 \, cm^3 \cdot 298.15 \, K}{30.5 \, cm^3} \]

### Step 5: Calculate \( T_2 \)
Performing the calculation:
\[ T_2 = \frac{45.8 \cdot 298.15}{30.5} \approx 455.15 \, K \]

### Step 6: Convert \( T_2 \) back to Celsius
To convert back to Celsius:
\[ T_2(°C) = T_2(K) - 273.15 \]
\[ T_2(°C) = 455.15 - 273.15 \approx 182.0°C \]

### Final Answer
The highest temperature to which the cylinder can be heated at constant pressure without having the piston come out is approximately **182.0°C**.