For a closed thermodynamic system of constant composition, specify the natural variables of the following state functions and write down an equation for the complete differential of the state function with respect to the natural variables. (i) internal energy u. enthalpy h. helmholtz free energy f. [1 mark] [1 mark] [1 mark] (iii) (b) using the first and second laws of thermodynamics, the equations from part (a), equations relating enthalpy and, separately, helmholtz free energy to internal energy and the maxwell relation for helmholtz free energy, derive an expression for the partial derivative of enthalpy with respect to volume at constant temperature in terms of temperature, volume and pressure. [5 marks] using the result from part (b), explain which physical properties of a pure substance should be measured to calculate the change in enthalpy with a change in volume at constant temperature.

Amass m = 74 kg slides on a frictionless track that has a drop, followed by a loop-the-loop with radius r = 18.4 m and finally a flat straight section at the same height as the center of the loop (18.4 m off the ground). since the mass would not make it around the loop if released from the height of the top of the loop (do you know why? ) it must be released above the top of the loop-the-loop height. (assume the mass never leaves the smooth track at any point on its path.) 1. what is the minimum speed the block must have at the top of the loop to make it around the loop-the-loop without leaving the track? 2. what height above the ground must the mass begin to make it around the loop-the-loop? 3. if the mass has just enough speed to make it around the loop without leaving the track, what will its speed be at the bottom of the loop? 4. if the mass has just enough speed to make it around the loop without leaving the track, what is its speed at the final flat level (18.4 m off the ground)? 5. now a spring with spring constant k = 15600 n/m is used on the final flat surface to stop the mass. how far does the spring compress?