6. For each of the following cases, calculate q, w, AS, and ASsur, ASTotal and determine if the process is spontaneous. In each case, assume that the process involves one mole ofan ideal gas with Cr 3R is initially at 300 K and 1 atm. a) The gas is heated at constant pressure to a final temperature of 450 K. Assume that the surroundings are at a temperature of 450 K and pressure of 1 atm throughout the process. b) The gas is heated at constant volume corresponding to the initial volume to final temperature of 450 K. Assume that the surroundings are maintained at temperature of 450 K and pressure of 1 atm throughout the process. c) The gas undergoes a reversible isothermal expansion at 300 K until the pressure is half its initial value. 7. a) Starting with the fundamental equation for the internal energy that incorporates both the first and second laws of thermodynamics and using the definitions of the remaining state functions, derive the differential expressions for U, H. A, and G and identify the natural variables for each of these functions. b) Use these four differential expressions to derive eight expressions that relate partial derivatives of the four energy functions with respect to their natural variables to a thermodynamic variable c) Use the fact that these functions are state functions to derive the four Maxwell relations 8. Use the fact that S is a state function to derive the thermodynamic equation of state. ap CT