1. Suppose f, g, h are functions of x. f = f(x), g = g(x), and h = h(x). Write down expressions for
a. d/dx (f + g + h)
b. d/dx fg
c. d/dx f( g + h)
2. Suppose v = v(P, T ). Write down expressions for
a. ∂/∂T 1/v
b. ∂/∂p 1/v
c. ∂/∂T ∂v/∂P
d. ∂/∂P ∂v/∂T
e. ∂/∂T (1/v ∂v/∂P)
f. ∂/∂P (1/v ∂v/∂T)
3. (n-dimensional heat kernel) Consider the heat kernel, h(x,t) = 1/tn/2 e-x2/4t and compute the derivatives
a. ∂h/∂t
b. ∂h/∂x
c. ∂2h/∂t
d. ∂2h/∂t
e. ∂2h/∂t
Hint: Be smart about simplifying each expression as you work & look for patterns.
4. Find as much information as you can about the function, f, given that ∇f(x, y, z) = (x, y, z)
5. Systems 1, 2, and 3 are gases with coordinates P1, V1, P2, V2, and P3, V3, respectively. When systems 1 and 3 are in equilibrium
P1V1 - nbP1 - P3V3 = 0
and when systems 2 and 3 are in equilibrium,
P2V2 - P3V3 + ncP3V3/V2 = 0
where n, c, and b are constants.
a. what are the three functions which are equal to one another at thermal equilibrium and each of which is equal to T, empiric temperature (on some scale).
b. What is the relation expressing thermal equilibrium between systems 1 and 2?
6. For an ideal gas show that
a. β = 1/T
b. k = 1/P
7. An approximate equation of state of a real gas at moderate pressure is given by
Pv = RT(1 + b/v) where b = b(T). Show that
a. β = 1/T v+ b + Tb/ v+2b
b. k = 1/P 1/ 1 + bRT/Pv2
8. At the critical point (∂P/∂T)T = 0. Show that both volume expansivity and compressibility approach infinity at the critical point.
9. During a quasi-static and adiabatic expansion of an ideal gas, the pressure at any moment is given by
PVy = K
where y and K are constants.
a. Show the work done by the gas during expansion from initial state (Pi, Vi), to final state (Pf, Vf) is given by
W = PiVi -PfVf/y-1
b. Assume the initial pressure and volume are 10 atm and 1 liter, respectively, and the final pressure and volume are 2 atm and 3.16 liters. How much work is done by a gas whose y = 1.4 = 7/5?