The raise in the temperature of a given mass of an ideal gas at constant pressure and at temperature `27^@` to double its volume is

Options

A. `327^@ C`
B. `54^@ C`
C. `300^@ C`
D. `600^@ C`

Answer

To solve the problem of determining the raise in temperature of a given mass of an ideal gas at constant pressure when its volume is doubled, we can follow these steps:

### Step 1: Understand the relationship between pressure, volume, and temperature.
According to the ideal gas law, we have the equation:
\[ PV = nRT \]
Where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles (constant for a given mass)
- \( R \) = universal gas constant (constant)
- \( T \) = temperature in Kelvin

Since the pressure is constant, we can express the relationship between volume and temperature as:
\[ \frac{V_1}{V_2} = \frac{T_1}{T_2} \]

### Step 2: Define the initial conditions.
Let:
- Initial volume \( V_1 = V_0 \)
- Final volume \( V_2 = 2V_0 \)
- Initial temperature \( T_1 = 27^\circ C = 27 + 273 = 300 \, K \)

### Step 3: Set up the equation using the volume and temperature relationship.
Using the relationship derived earlier:
\[ \frac{V_0}{2V_0} = \frac{T_1}{T_2} \]
This simplifies to:
\[ \frac{1}{2} = \frac{300}{T_2} \]

### Step 4: Solve for the final temperature \( T_2 \).
Cross-multiplying gives:
\[ T_2 = 2 \times 300 = 600 \, K \]

### Step 5: Convert the final temperature back to Celsius.
To convert from Kelvin to Celsius:
\[ T_2 = 600 \, K - 273 = 327^\circ C \]

### Step 6: Calculate the raise in temperature.
The raise in temperature \( \Delta T \) is given by:
\[ \Delta T = T_2 - T_1 = 327^\circ C - 27^\circ C = 300^\circ C \]

### Final Answer:
The raise in temperature of the gas to double its volume at constant pressure is \( 300^\circ C \).

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