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1. Starting with internal energy U, entropy S, volume V and mole number N (for a single-componenent system), define, in succession, the following quantities: temperature T, pressure P, chemical potential , specific Gibbs free energy g, residual Gibbs free energy gR, and fugacity f 2. Given the fundamental relation S=NRlog[(Nu0U)cNv0V] where R is the ideal gas constant, and u0 and v0 are constants, find the predicted PvT equation of state for this model, and the chemical potential as a function of temperature and pressure. What is the fugacity? 3. Consider two vectors: v1:=2x+3xz and v2:=x+2x+3z. Find the following quantities: (a) v1 (b) v1v2 (c) The angle between the two vectors (in radians). (d) The angle that v2 makes with the z-axis