### Homework 4

```CYL100 Physical Chemistry Homework 3
Handed out: Feb. 19, 2014
Due: Feb. 25, 2014
You will find any data that is not provided on the back pages of Atkins’. Consider the
system to be an ideal gas, if it is not mentioned but required.
1. At high densities the molecules themselves make up an appreciable fraction of the
available volume, and an equation of state of 1 mol of gas can be written in the form
P(V − b) = RT, where ’b’ is a constant which is related to the size of the molecules.
(i)Calculate the work done when 1 mole of this nonideal gas at 298 K is compressed
reversibly and isothermally from 22.4 L to 0.224 L if ’b’ is 0.04L. (ii) What is the work
done if the gas behaves ideally in (i) and explain why the answers differ in (i) and (ii).
2. Verify that the enthalpy of a perfect gas is independent of its pressure while it is
dependent upon pressure for a van der Waals gas. [Use the approximate form of van der
Waals equation PV = nRT + nPb − ( a/RT )]
3. An important application of adiabatic cooling/heating is in atmospheric physics. Pressure
varies with altitude as dP/dh = − gρ, where ρ is the density. Calculate the change in
temperature at the top of a mountain 2 km above the valley floor assuming the process
is adiabatic and air behaves ideally.
√
4. The velocity of sound in a fluid is given as c = 1/ ρκS , where κS , the adiabatic com pressibility is given as −1/V ∂V
. What is the velocity of sound in an ideal gas?
∂p
S
of these
5. Which
H
H cyclic integrals Hmust vanish
H for a closed
H system Hwith P-V work only? (a)
PdV; (b) ( PdV + VdP); (c) VdV; (d) HdT; (e) dqrev ; (f) dwrev /P.
6. One mol of hydrogen occupies a volume of 0.1 m3 at 300K and one mol of argon also
occupies the same volume but at 400K. While isolated from their surroundings, each
undergoes a free expansion, the hydrogen to 5 times and argon 8 times its initial volume.
The two masses are then placed in contact with each other and reach thermal equilibrium.
What is the total change in entropy? (CvH2 = 10 kJ/K/kg, CvAr = 0.31 kJ/K/kg)
7. The molar heat capacity of oxygen at constant pressure for temperatures in the range
300 to 400 K and for low or moderate pressures can be approximated as CP (J mol−1 K−1 )
= 25.73 + 0.013 T. (a) Calculate q, w, ∆U, ∆H when 2 mol of O2 is reversibly heated from
27 to 127 °C with P held fixed at 1 atm. (b) Repeat the calculation if the same change is
effected at constant V.
8. By a thermodynamic analysis show that the following familiar processes are spontaneous:
(a) A book is pushed off a table and falls to the floor. (b) One mole of an ideal gas in a
vessel that is connected to another identical evacuated vessel through a valve. The valve
is opened and the gas occupies both vessels. (c) Melting of ice when brought in contact
with an object at a temperature above 0 °C.
9. 100 g of ice at 0 °C is dropped into an insulated beaker containing 150 g of water at 100
°C. Calculate ∆S for this process.
10. Calculate the maximum work and the maximum non-expansion work that can be
obtained from the freezing of supercooled water at −5 °C and 1.0 atm. The densities of
water and ice are 0.999 and 0.917 g cm−3 , respectively at −5 °C.
∂V 2
∂P
11. Derive the relations: (i) C p − Cv = T ∂T
; (ii) C p − Cv = α βTV ; (iii) µ JT =
V ∂T P
CV β
∂V
−(V/C p )(βCv µ J − βP + 1) (iv) ∂H
=
γ/β
(v)
∂V S
∂T P = Tα .
12. At what pressure can graphite be converted to diamond. Use the data given below.
∆H0298 kJ mol −1 S0298 J K −1 mol −1 ρ298 g cm−3
graphite
0
5.69
2.22
diamond
1.9
2.44
3.51
```