At `27^(@)C`, a ges is compressed to half of its volume . To what temperature it must now be heated so that gas occupies just its original volume ?

Options

A. `54^(@)C`
B. `600^(@)C`
C. `327^(@)C`
D. `327K`

Explanation

To solve the problem, we will use the relationship between the volume and temperature of a gas, which is derived from the ideal gas law. Here's a step-by-step solution:

### Step 1: Convert the initial temperature from Celsius to Kelvin
The initial temperature is given as \(27^\circ C\). To convert this to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273 \]
Thus,
\[ T_1 = 27 + 273 = 300 \, K \]

**Hint:** Remember to always convert Celsius to Kelvin when using gas laws.

### Step 2: Define the initial and final volumes
Let the initial volume of the gas be \(V\). When the gas is compressed to half of its volume, the new volume \(V_1\) is:
\[ V_1 = \frac{V}{2} \]

**Hint:** Identify the initial and final states of the gas clearly.

### Step 3: Set up the relationship between volume and temperature
According to Charles's Law, the volume of a gas is directly proportional to its temperature when pressure is constant. This can be expressed as:
\[ \frac{V_1}{V_2} = \frac{T_1}{T_2} \]
where:
- \(V_1\) is the volume after compression (\(\frac{V}{2}\)),
- \(V_2\) is the original volume (\(V\)),
- \(T_1\) is the initial temperature (300 K),
- \(T_2\) is the temperature we need to find.

**Hint:** Recall that Charles's Law relates volume and temperature.

### Step 4: Substitute the known values into the equation
Substituting the known values into the equation:
\[ \frac{\frac{V}{2}}{V} = \frac{300}{T_2} \]
This simplifies to:
\[ \frac{1}{2} = \frac{300}{T_2} \]

**Hint:** Simplifying the equation can help isolate the variable you need to find.

### Step 5: Solve for \(T_2\)
Cross-multiplying gives:
\[ T_2 = 300 \times 2 = 600 \, K \]

**Hint:** Be careful with algebraic manipulation to avoid mistakes.

### Step 6: Convert the final temperature back to Celsius
To convert \(T_2\) back to Celsius:
\[ T(°C) = T(K) - 273 \]
Thus,
\[ T_2(°C) = 600 - 273 = 327 \, °C \]

**Hint:** Always convert back to Celsius if the question requires it.

### Final Answer
The gas must be heated to \(327^\circ C\) to occupy its original volume.

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