Gas molecules have three degrees of freedom for translational motion along the x, y, and z directions. Additionally, for a diatomic gas molecule, there are two more degrees of freedom for rotational motion. Therefore, the total number of degrees of freedom for a diatomic gas molecule is five.

For an ideal gas, the internal energy depends only on temperature and is given by the equation:

U = (f/2) * N * k * T

where U is the internal energy, f is the number of degrees of freedom, N is the number of gas molecules, k is the Boltzmann constant, and T is the temperature.

Since the gas is expanding adiabatically, there is no heat exchange with the surroundings, and the process is reversible. In an adiabatic expansion, the relationship between pressure (P), volume (V), and temperature (T) is given by:

PV^(gamma) = constant

where gamma is the ratio of specific heat capacities (Cp/Cv) for the gas.

In this case, the volume increases to 5.66V and the temperature decreases. Therefore, using the adiabatic expansion formula, we can write:

(P * V^gamma) / T = constant

Since the initial and final states of the gas are known, we can use the above equation to find the new temperature (T') at the final volume (5.66V).

Now, let's consider the initial and final states of the gas:

Initial state:
Pressure = P
Volume = V
Temperature = T

Final state:
Pressure = ?
Volume = 5.66V
Temperature = T'

Using the adiabatic expansion formula, we can write:

(P * V^gamma) / T = (P' * (5.66V)^gamma) / T'

Since the process is adiabatic, the initial and final pressures can be related using the ideal gas law equation:

P * V = P' * (5.66V)

Simplifying the equation:

P' = (P * V) / (5.66V)

P' = P / 5.66

Substituting this value of P' into the adiabatic expansion formula:

(P * V^gamma) / T = (P / 5.66 * (5.66V)^gamma) / T'

Simplifying the equation:

V^gamma / T = (1 / 5.66 * (5.66V)^gamma) / T'

V^gamma / T = V^gamma / (5.66 * T')

Since the initial and final volumes are not equal to zero, we can cancel out V^gamma from both sides of the equation:

1 / T = 1 / (5.66 * T')

Solving for T':

T' = 5.66 * T

Therefore, the final temperature is 5.66 times the initial temperature.

Since the temperature has changed, but the gas molecules have not gained or lost any degrees of freedom, the number of degrees of freedom for the gas molecules remains the same. Thus, the gas molecules still have five degrees of freedom.

Therefore, the correct answer is option C) 5.