problem 1: Heat capacities C_v and C_p are defined as temperature derivatives respectively of U and H. Show that the general expression connecting C_p to C_v is:

C_p = C_v  + T(∂P/∂T)_V (∂V/∂T)_P

problem 2: Liquid isobutane is throttled through a valve from an initial state of 360 K and 4000 kPa to a final pressure of 2000 kPa. Estimate the entropy change of the isobutane. The specific heat of liquid isobutane at 360 K is 2.78 J/g/°C. For volume calculation, use equation:

V = V_cZ_c^{(1-T_r)^0.2857}

Given: V_c = 262.7 cm³/mol, Z_c= 0.282 and T_c= 408.1 K

Also, change in T and V is negligible during throttling process.

problem 3: A concentrated binary solution containing mostly species 2 (but x₂ not equals to 1) is in equilibrium with a vapor phase containing both species 1 and 2. The pressure of this two-phase system is 1 bar; and the temperature is 25°C. Determine from the following data good estimates of x₁ and y₁.

Given: H₁ = 200 bar, P₂^sat = 0.1 bar

State and justify all assumptions.

problem 4: For the acetone(1)/methanol(2) system a vapor mixture for which z₁ = 0.25 and z₂ = 0.75 is cooled to temperature T in the two-phase region and flows into a separation chamber at a pressure of 100 kPa. If the composition of the liquid product is to be x₁= 0.175, what is the required T and what is the value of y₁. Also find out the molar fraction of the system in the vapor phase.

Given:

lnγ1 = 0.64 x₂²
lnγ2 = 0.64 x₁²

ln(P^sat) = A –  B/(C + T(°C))

Where, for Acetone: A = 14.3916; B = 2795.82; C = 230.00

for Methanol: A = 16.5938; B = 3644.30; C = 239.76