One mole of an ideal gas with `gamma`=1.4, is adiabatically compressed so that its temperature rises from `27^@C` to `35^@C`. The change in the internal energy of the gas is (R = 8.3J/mol.K)

Options

A. -166 J
B. 166 J
C. -168 J
D. 168 J

Explanation

To solve the problem of finding the change in internal energy of one mole of an ideal gas undergoing adiabatic compression, we can follow these steps:

### Step 1: Understand the relationship for internal energy change
For an ideal gas, the change in internal energy (\( \Delta U \)) during an adiabatic process can be expressed as:
\[ \Delta U = nC_V \Delta T \]
Where:
- \( n \) = number of moles of gas (1 mole in this case)
- \( C_V \) = molar specific heat at constant volume
- \( \Delta T \) = change in temperature

### Step 2: Calculate the change in temperature
Convert the temperatures from Celsius to Kelvin:
- Initial temperature \( T_1 = 27^\circ C = 27 + 273 = 300 \, K \)
- Final temperature \( T_2 = 35^\circ C = 35 + 273 = 308 \, K \)

Now, calculate the change in temperature:
\[ \Delta T = T_2 - T_1 = 308 \, K - 300 \, K = 8 \, K \]

### Step 3: Determine \( C_V \) using \( \gamma \)
The relationship between \( C_V \) and \( R \) (the universal gas constant) is given by:
\[ C_V = \frac{R}{\gamma - 1} \]
Given \( R = 8.3 \, J/(mol \cdot K) \) and \( \gamma = 1.4 \):
\[ C_V = \frac{8.3}{1.4 - 1} = \frac{8.3}{0.4} = 20.75 \, J/(mol \cdot K) \]

### Step 4: Calculate the change in internal energy
Now we can substitute the values into the equation for \( \Delta U \):
\[ \Delta U = nC_V \Delta T = 1 \cdot 20.75 \cdot 8 = 166 \, J \]

### Conclusion
The change in internal energy of the gas is:
\[ \Delta U = 166 \, J \]