1. A measure of a medium s stiffness (it s resistance to compression) is given in terms of the bulk modulus, B, defined by

Transcription

1 1. A measure of a medium s stiffness (it s resistance to compression) is given in terms of the bulk modulus, B, defined by B V P V a) Assuming isothermal compression, derive a formula for the bulk modulus of an ideal gas as a function of pressure. b) Assuming adiabatic compression, derive a formula for the bulk modulus of an ideal gas as a function of pressure. c) The speed of sound is c s = B/ρ in which ρ is the density of a gas. How does the speed of sound change with the vertical, z, as the sound wave moves vertically upward in a stationary isothermal atmosphere of temperature T 0?

2 2. In the examination of a two-state paramagnet with N up and N down dipoles, we found that the multiplicity function for having N dipoles amongst the N = N +N dipoles was ( N Ω = N ) N 1/2 2 N exp ( (N N/2) 2 ) 2σ 2 in which σ = N/2 is the standard deviation, and the approximation assumes N is large and N N/2. Use this result, by analogy, to examine the spread of a pollutant in air modelled as a one dimensional random walk. Assume a particle is situated at x = 0 at time t = 0. At each time step t, the particle moves randomly either forward or backward a distance x. a) Where is the particle most likely situated after a time t = N t, with N large? b) The particle has a high probability of being located within one standard deviation of its most likely location. Find a formula for the standard deviation of the position, x, as a function of t, x and t. c) Now suppose the particle is a molecule of hydrogen sulfide, H 2 S. It travels at v = 500 m/s colliding every t = s. After a collision assume the particle randomly either continues in its direction or it reverses its direction. Using your result in b), estimate how far it is likely to have moved from its start position after 1 second and 1 hour. (Give your answer in meters.)

3 3. Suppose an ideal diatomic gas operates in a heat engine whose rectangular PV cycle is shown to the right. a) Evaluate the efficiency of this engine for the case V 2 = 3V 1 and P 2 = 2P 1. b) Calculate the efficiency of an ideal (i.e. Carnot) engine operating between the same temperature extremes. (In both a) and b) express your answer as a fraction of two integers.) P P 2 P 1 B C A D V 1 V 2 V

4 4. Assume that the air you exhale is at T B = 35 C, with a relative humidity of 90%. Also suppose that V B = 1 litre of your warm breath mixes with V E = 10 litres of environmental air at T E = 10 C. You are able to see your breath if cloud droplets form as your breath cools to 10 C and combines with the moisture in the environment. a) Calculate the saturation vapour pressure, e se, for air at temperature T E. b) Calculate the partial vapour pressure, e B, associated with moisture in your breath. c) Assuming the mass, m B, of vapour in your breath is small compared to the mass of air, write a formula for m B in terms of V B, e B, the molar mass ratio of water to air (ǫ w ) and the air density (ρ) and pressure (p). d) Likewise write a formula for the unknown mass, m E, of vapour in the environment having partial vapour pressure e E. Combine this with your result in c) to write an expression for the partial vapour pressure, e T, of the breathenvironment mixture. (Assume the change in volume of your cooling breath is negligible and write the formula in terms of V B, e B, V E and e E.) e) Assuming the mixture is just at saturation (so you can see your breath), use your result in d) and the values you computed in a) and b) to find the relative humidity (in percent) of the environment before you breathed into it. (In this problem you may assume the partial vapour pressure equation for e s (T) with e 0 = 611Pa and T 0 = 273K is valid for temperatures up to 35 C).

5 5. The energy levels of a harmonic oscillator in an Einstein solid are 0, hω, 2 hω,..., in which ω is a constant and h = h/2π, with h being Planck s constant. a) Write down the partition function for a system of N identical harmonic oscillators, simplifying your result using the formula for a geometric series: 1+x+x = 1/(1 x). b) Use this result to find the average energy of the system in the high temperature limit.

6 6. The energy density of photons in a box (a photon gas ) at temperature T is u = 4 c σt4 in which c is the speed of light and σ is the Stefan-Boltzmann constant. a) What is the heat capacity of photons in a box of volume V? (Give your answer in terms of T, V, c and σ.) b) Assuming this formula is accurate all the way down to absolute zero, what is the entropy of the photon gas, with S(T) defined so that S(0) = 0? c) Use P = ( ) U V to find the pressure of a photon gas as a function of the energy density, u. S,N