## 5.6 Answers to the Problem Set

1. Definitions are in the text.
1. All false.
#a)
#temperatures in C, °F = 1.8°C + 32
temps.C= c(46, 30, 15, 0, -10, -23)
#Convert to F
1.8*temps.C+32
##  114.8  86.0  59.0  32.0  14.0  -9.4
#Convert to K, °K = C + 273.15
temps.C+273.15
##  319.15 303.15 288.15 273.15 263.15 250.15
#b)
temps.K=c(50, 300, 350)
#Convert to C
temps.K-273.15
##  -223.15   26.85   76.85
temps.F= 72
#Convert to C, °C = 5/9 ( F - 32)
5/9*(temps.F-32)
##  22.22222
#c)
temps.F= c(-66,134)
#To C
5/9*(temps.F-32)
##  -54.44444  56.66667
#d)
#x=1.8*x+32
#-0.8x=32
32/-0.8
##  -40
1. $$W=\int PdV=\int \frac{F}{A}\cdot A\cdot dl=\int Fdl$$ where $$F$$ is force, $$A$$ is area and $$dl$$ is the change in length.

\begin{align*} E_p &= mgh = 2 kg \cdot 9.8 m s^{-2}\cdot 50 m = 980 kg m^2 s^{-2} = 980 J \\ E_k &= \frac{1}{2}mv^2 = \frac{1}{2} \cdot 2 kg (2 m s^{-1})^2 = 4 J \\ \end{align*}

1. $$\Delta E_p + \Delta E_k + \Delta U = 0$$

2. All the potential energy is now kinetic energy which now totals 984 J.

3. Turbulent mixing occurs when the water hits. The temperature rise is $\frac{984}{2 kg\cdot 4184Jkg^{-1}°C^{-1}}=0.12°C$ 5

$$\Delta U$$ $$W$$ $$Q$$
adiabatic $$-W$$ $$C_VdT$$ 0
constant temperature 0 $$RT\:ln\frac{P_1}{P_2}$$ $$W$$
constant volume $$Q$$ 0 $$C_VdT$$
constant pressure $$C_VdT$$ $$P\Delta V$$ $$W+\Delta U$$
1. Class discussion

1. This is an ongoing area of research. Hamilton (1973) used a general comparative approach and found the hotter is faster up to about 40°C for many terrestrial organisms. Eppley (1972) found a similar result for phytoplankton. Calloway (1975) has suggested this is due to the thermodynamic properties of liquid water, the medium in which life processes are carried out. [Many more recent examples are available.]
1. Having a higher body temperature implies faster growth and response to environmental stimuli. The homeothermic animals must process a great deal more energy for the same amount of growth but being “hot-blooded” they are more independent from the physical environment.

2. The First Law is the basis for describing energy relations from the molecular to the ecosystem level. Since life depends on degrading energy (Second Law) one would expect that thermodynamics might yield important insights into ecological relationships and the nature of life itself.