Given:
- Heat absorbed by the gas: 1831.5 J
- Work done by the gas: 1000 J
- Initial temperature: 27°C

Formula:
The first law of thermodynamics states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. Mathematically, it can be represented as:
ΔU = Q - W

Where:
ΔU = Change in internal energy
Q = Heat added to the system
W = Work done by the system

Calculations:
1. Convert the initial temperature from Celsius to Kelvin:
T1 = 27°C + 273.15 = 300.15 K

2. Since the gas is diatomic, its number of degrees of freedom is 5 (3 translational + 2 rotational).

3. The change in internal energy can be calculated using the formula:
ΔU = (5/2) nR ΔT

Where:
n = Number of moles of gas
R = Ideal gas constant (8.314 J/mol·K)
ΔT = Change in temperature in Kelvin

4. The work done by the gas can be calculated using the formula:
W = -PΔV

Since the gas is ideal, we can use the formula for an ideal gas:
W = -nRT ln(V2/V1)

Where:
P = Pressure
V1 = Initial volume
V2 = Final volume

Since the volume is not given, we cannot calculate the work done by the gas.

5. Rearranging the first law of thermodynamics equation, we can solve for the change in temperature:
ΔT = (2ΔU)/(5nR)

6. Substitute the given values into the equation to find the change in temperature:
ΔT = (2 * 1831.5 J) / (5 * 1 mol * 8.314 J/mol·K) = 43.97 K

7. Convert the change in temperature from Kelvin to Celsius:
ΔT = 43.97 K - 273.15 = -229.18°C

Answer:
The change in temperature observed is -229.18°C, which is not one of the given options (A, B, C, or D).

Is the option is b