The change in internal energy, when a gas is cooled from $927^{\circ} \mathrm{C}\( to \)27^{\circ} \mathrm{C}$ (A) \(200 \%\) (B) \(100 \%\) (C) \(300 \%\) (D) \(400 \%\)

To work with temperatures in thermodynamics, we need to convert the given temperatures from Celsius to Kelvin. The conversion formula is: \(K = ^\circ C + 273.15\) For the initial temperature (927°C): \(T1 = 927 + 273.15 = 1200.15 K\) For the final temperature (27°C): \(T2 = 27 + 273.15 = 300.15 K\)

The internal energy of an ideal gas depends solely on its temperature. The change in internal energy is proportional to the change in temperature. So, the ratio of final to initial internal energy can be given as: \(\frac{U2}{U1} = \frac{T2}{T1}\) Using the temperatures calculated in Step 1: \(\frac{U2}{U1} = \frac{300.15}{1200.15}\)

To find the percentage change in internal energy, we can use the following formula: Percentage Change = \(\frac{U2 - U1}{U1} * 100\%\) First, we need to find the absolute change in internal energy, which can be calculated using the ratio we found in Step 2: \(U2 - U1 = U1 * (\frac{U2}{U1} - 1)\) Now, divide the absolute change in internal energy by the initial internal energy and multiply by 100 to find the percentage change: Percentage Change = \(\frac{U1 * (\frac{U2}{U1} - 1)}{U1} * 100\%\) After simplifying the expression, we get: Percentage Change = \((\frac{U2}{U1} - 1) * 100\%\) Using the ratio calculated in Step 2: Percentage Change = \((\frac{300.15}{1200.15} - 1) * 100\% \approx -75 \% \) So, the percentage change in internal energy is approximately -75%. None of the given options match this result, indicating that there might be an error in the question or choices provided.