Tipler Chapter 20
Name: ________________________
#1 Expansion
When the temperature of a nickel coin is raised from 0 oC to 160 oC, the radius increases by 0.208 %.
a) What is the percent increase in the area of the face of the coin?
b) What is the percent increase in the thickness of the coin?
c) What is the percent increase in the volume of the coin?
d) What is the percent increase in the mass of the coin?
e) What is the coefficient of linear expansion for the coin?
f) What is the coefficient of volume expansion for the coin?
#2 Expansivity & Compressibility
Recall that the volume expansion coefficient, , relates the fractional change in volume, (V/V), and the
change in temperature, T as follows:
Δ𝑉
= 𝛽Δ𝑇
𝑉
Upon rearranging terms, the volume expansion coefficient (or expansivity as it is sometime called) can
be written:
1 Δ𝑉
𝛽= ( )
𝑉 Δ𝑇
In many circumstances (such as with gases), the expansivity is NOT a constant. That is, the same gas can
have a different change in volume for a given change in temperature depending on what initial
temperature is chosen. Thus, in order to find the value of the expansivity of a gas at a particular
temperature, the temperature interval must be made infinitesimally small. In other words, take a
derivative. Taking the limit as T goes to zero,
1 ∂𝑉
𝛽= ( )
𝑉 ∂𝑇
Here, the partial derivative is taken (rather than a total derivative) to imply that the volume is changing
only due to changes in temperature. That is, pressure, P, is a constant.
(a)Starting with the equation of state for an ideal gas, PV = nRT, calculate the expansivity of an ideal gas
at a temperature of 300 K.
(b) Now consider the change in volume of an ideal gas when the pressure is changed while the
temperature is held constant. The compressibility of a material (denoted by ) is defined in a similar
way as the expansivity. That is, the compressibility is defined as:
1 𝜕𝑉
𝜅= − ( )
𝑉 ∂𝑃
The minus sign is included because an increase in pressure always produces a decrease in volume
under constant temperature, so (𝜕𝑉/𝜕𝑃) is inherently negative.
Proceeding in a way similar to part (a), calculate the compressibility of an ideal gas at a pressure of 1.0 x
105 Pa.
#3 Conductive Heat Transfer
During a cold Chicago winter, a steady thermodynamic state has been reached in a small pond with a
layer of ice on the top. The air directly above the pond is at –8.0 oC and the ground at the bottom of the
pond is held at 4.0 oC. The pond holds a total of 450 m3 of water and has a relatively uniform depth
(water and ice) of 1.5 m. Assume the thermal conductivities of ice and water are 0.40 cal/m/s/Co and
0.12 cal/m/s/Co, respectively. Calculate the thickness of the ice.
#4 Hot Iron
A clothes iron with a rating of 1100 W has a hot surface area of 400 cm2 and an emissivity equal to 0.97.
The surface temperature of the iron is 185 oC and the room where the iron is sitting is at a temperature
of 27 oC.
(a) At what rate is the hot iron <i>emitting</i> energy into its surroundings?
(b) At what rate is iron <i>absorbing</i> energy from its surroundings?
(c) What is the <i>net rate</i> at which energy is transferred from the iron by <i>radiation</i>?
(d) How much energy is transferred by radiation in 6.0 minutes?
(e) What is the net rate at which energy is transferred from the iron by conduction and convection?
(f) How much energy is transferred by conduction and convection in 6.0 minutes?