`4.48 L` of an ideal gas at `STP` requires 12 cal to raise its temperature by `15^(@)C` at constant volume. The `C_(P)` of the gas is

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Determine the number of moles of the gas
We know that 1 mole of an ideal gas occupies 22.4 L at STP (Standard Temperature and Pressure). Given that we have 4.48 L of the gas, we can find the number of moles (n) using the formula:

\[ n = \frac{\text{Volume of gas}}{\text{Molar volume at STP}} = \frac{4.48 \, \text{L}}{22.4 \, \text{L/mol}} \]

Calculating this gives:

\[ n = \frac{4.48}{22.4} = 0.2 \, \text{moles} \]

### Step 2: Calculate the specific heat at constant volume (C_V)
We are given that 12 calories are required to raise the temperature of 0.2 moles of the gas by 15°C. We need to find out how much energy is required to raise the temperature of 1 mole of gas by 1°C (C_V).

Using the formula:

\[ C_V = \frac{\text{Energy required}}{\text{Number of moles} \times \Delta T} \]

Substituting the values:

\[ C_V = \frac{12 \, \text{cal}}{0.2 \, \text{moles} \times 15 \, \text{°C}} = \frac{12}{3} = 4 \, \text{calories} \]

### Step 3: Use the relationship between C_P and C_V
For an ideal gas, the relationship between the specific heats at constant pressure (C_P) and constant volume (C_V) is given by:

\[ C_P - C_V = R \]

Where R is the gas constant. In terms of calories, R is approximately 2 cal/mol·°C.

### Step 4: Calculate C_P
Now, we can find C_P using the values we have:

\[ C_P = C_V + R \]

Substituting the known values:

\[ C_P = 4 \, \text{cal} + 2 \, \text{cal} = 6 \, \text{calories} \]

### Final Answer
The specific heat at constant pressure (C_P) of the gas is **6 calories**.

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