A sample of gas is found to occupy a volume of `900 cm^(3)` at `27^(@)C`. Calculate the temperature at which it will occupy a volume of `300 cm^(3)`, provided the pressure is kept constant.

Explanation

To solve the problem, we will use Charles's Law, which states that at constant pressure, the volume of a gas is directly proportional to its temperature in Kelvin. The formula for Charles's Law is:

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

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

### Step 1: Identify the given values
- Initial volume, \( V_1 = 900 \, \text{cm}^3 \)
- Initial temperature, \( T_1 = 27^\circ C \)
- Final volume, \( V_2 = 300 \, \text{cm}^3 \)

### Step 2: Convert the initial temperature from Celsius to Kelvin
To convert Celsius to Kelvin, use the formula:
\[ T(K) = T(°C) + 273 \]
So,
\[ T_1 = 27 + 273 = 300 \, K \]

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

### Step 4: Substitute the known values into the equation
Now substitute \( V_2 = 300 \, \text{cm}^3 \), \( T_1 = 300 \, K \), and \( V_1 = 900 \, \text{cm}^3 \):
\[ T_2 = \frac{300 \, \text{cm}^3 \cdot 300 \, K}{900 \, \text{cm}^3} \]

### Step 5: Calculate \( T_2 \)
\[ T_2 = \frac{90000 \, \text{cm}^3 \cdot K}{900 \, \text{cm}^3} = 100 \, K \]

### Step 6: Convert \( T_2 \) from Kelvin to Celsius
To convert Kelvin back to Celsius, use the formula:
\[ T(°C) = T(K) - 273 \]
So,
\[ T_2 = 100 - 273 = -173^\circ C \]

### Final Answer
The temperature at which the gas will occupy a volume of \( 300 \, \text{cm}^3 \) is \( -173^\circ C \).

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