The isobaric expansion coefficient of any system quantifies how the system expands as its
temperature increases and is
:=
1
V
!V
T
P
.The isothermal compressibility quantifies how the system expands as its pressure increases
and is
T := 1
V
!V
P
T
.
a) Show that, for any system
T
=
!P
T
V
and check this for an ideal gas.These can be used to check how the pressure must increase when the temperature is raised
and while the volume of an object is held constant since
dP =
!P
T
V
dT
if V is constant.b) In a particular temperature and pressure range the compressibility of water is 4.52
1010 Pa1 and the expansion coefficient is 2.1 104 K1. Determine by how much
the pressure on the water must increase to increase it temperature by 10 C while the
volume is kept constant.c) In a particular temperature and pressure range the compressibility of gold is 4.5
1012 Pa1 and the expansion coefficient is 0.42 104 K1. Determine by how much
the pressure on the gold must increase to increase it temperature by 10 C while the
volume is kept constant.d) Explain why it is more difficult to raise the temperature of water and gold by the same
amount at constant volume rather than to do so at constant pressure.A detailed analysis of vibrations in a material give that the speed of sound in a material is
vsound =
#
P
where P is the pressure of the material and the mass per unit volume.a) Show that for sound in any gas
vsound =
#
B
where the bulk modulus is
B = V P
Vb) Determine the bulk modulus for an ideal gas that undergoes an isothermal process and
use the result to determine an expression for the speed of sound if the sound propagation
occurs via an isothermal process.c) Show that for an adiabatic process
B = P
where = cP /cV . Use the result to determine an expression for the speed of sound if
the sound propagation occurs via an adiabatic process.d) How do the two speeds compare? Which process is more likely to occur for sound
propagation in air?e) Assume that air is an ideal gas determine a value for the speed of sound in air as
predicted by the above analysis.a) Determine an expression for the enthalpy in terms of T, P, and V for a monoatomic
ideal gas (this will require using the known expressions for internal energy). Use the
expression for enthalpy to determine expressions for cV and cP .b) Determine an expression for the enthalpy in terms of T, P, and V for a diatomic ideal gas.
Use this to determine expressions for cV and cP . For a diatomic ideal gas E = 5
2 P V .c) For carbon monoxide, measurements give that, provided that the temperatures are between 298.15 K and 1200 K, the enthalpy per mole is
H = H0 + AT +
B
2
T2 +
C
3
T3
where A = 25.57 J/mol K, B = 6.096 103 J/mol K2, C = 4.055 106 J/mol K3 and
T is the temperature in Kelvin (source: NIST Chemistry webbook). Determine an
expression for the heat capacity at constant pressure per mole for carbon monoxide.Determine the heat capacities at 300 K and 400 K? How do these compare to the heat
capacity for a diatomic ideal gas?d) Suppose that 1 mol of this gas is kept at constant pressure. Determine the heat needed
to raise the temperature from 300 K to 350 K. Determine the heat needed to raise the
temperature from 350 K to 400 K.e) Optional (up to 5 points extra credit): For any gas
cP =
!H
T
P
allows one to determine one partial derivative of H(T,P) from experimental measurements. By following a similar scheme as was done for energy, show how one can relate
the other partial derivative
!H
P
T
to cV and cP .A fire syringe consists of a cylinder containing a gas. The gas can be compressed rapidly
this process is adiabatic. Consider a cylinder the contains a diatomic ideal gas initially at
300 K (room temperature). Suppose that the initial volume of the cylinder is Vi and the final
volume is Vf .a) Determine an expression for the ratio of temperatures Tf /Ti in terms of Vi and Vf .
b) Choose a material that you would like to ignite and determine the compression ratio
Vf /Vi so that a fire syringe will ignite the material. Supply any relevant data needed to
answer the question.In the 2014 AFC Championship game, footballs were apparently deflated. NFL regulations
require that the pressure of the football be between 12.5 psi and 13.5 psi. It is possible that the
footballs could have been deflated during play as a result of the temperature drop between
room where they were inflated and the outside.Suppose that the balls were inflated to 13 psi
in a room at 75 F and then taken outside where the temperature was 51 F. Assume that
the air inside the ball was an ideal gas. Note that the stated pressures are gauge pressures,
i.e. pressure above atmospheric pressure. Also the game was played in Boston, MA, which is
at sea level.a) Suppose that the volume of the ball remained constant. Determine the pressure of the
air inside the ball once it is in equilibrium outside. In this process is any work done on
the ball? Does heat enter or leave the ball?b) Suppose that the ball were inflated from 0 psi to 13 psi. Assume that the air was initially
at room temperature before it was pumped into the ball and that the pumping process
was adiabtaic.Determine the temperature of the ball (in the room) immediately after
it had been pumped up. This ball is then taken outside. Determine the pressure in the
ball once it reaches equilibrium outside.c) Would either of these explain a reduction in pressure to 11.5 psi?
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