Internal energy of two moles of an ideal gas at a temperature of `127^(@)C is 1200R`. Then find the molar specific heat of the gas at constant pressure?

Explanation

To solve the problem, we need to find the molar specific heat at constant pressure (C_P) for a given ideal gas. The internal energy (U) of the gas is provided, and we can use the relationship between internal energy, specific heats, and the ideal gas law.

### Step-by-Step Solution:

1. **Identify Given Values**:
- Number of moles (n) = 2 moles
- Temperature (T) = 127°C = 127 + 273 = 400 K
- Internal energy (U) = 1200R

2. **Use the Formula for Internal Energy**:
The internal energy (U) of an ideal gas can be expressed as:
\[ U = n C_V \Delta T \]
where \(C_V\) is the molar specific heat at constant volume, and \(\Delta T\) is the change in temperature. However, since we have the internal energy directly, we can rearrange this to find \(C_V\):
\[ U = n C_V T \]
Rearranging gives:
\[ C_V = \frac{U}{n \Delta T} \]

3. **Substitute the Values**:
We know:
\[ U = 1200R, \quad n = 2, \quad T = 400 K \]
Thus:
\[ 1200R = 2 C_V (400) \]
Simplifying gives:
\[ C_V = \frac{1200R}{800} = 1.5R \]

4. **Find C_P Using the Relationship Between C_P and C_V**:
The relationship between the specific heats is given by:
\[ C_P = C_V + R \]
Substituting the value of \(C_V\):
\[ C_P = 1.5R + R = 2.5R \]

5. **Final Answer**:
The molar specific heat at constant pressure \(C_P\) is:
\[ C_P = 2.5R \]