A liquid has a density $\rho$.

Show that the fractional change in density for a change in temperature $\Delta T$ is

$$\Delta \rho / \rho=-\beta \Delta T$$

The theoretical lapse rate for dry air (no water vapor) in an atmosphere is given bywhere g is the acceleration due to gravity, M is the molar mass of the uniform ideal gas in the atmosphere, R is the gas constant, and γ is the ratio of molar specific heats, which we will study. (a) Calculate the theoretical lapse rate on the Earth given that γ = 1.40 and the effective molar mass of air is 28.9 g/mol. (b) Why is this value different from the value of –6.5$^ { \circ } \mathrm { C }$/km given in the text? (c) The atmosphere of Mars is mostly dry carbon dioxide, with a molar mass of 44.0 g/mol and a ratio of molar specific heats of γ = 1.30. The mass of Mars is $6.42 \times 1023$ kg and the radius is $3.37 \times 106$ m. What is the lapse rate for the Martian troposphere? (d) A typical surface atmospheric temperature on Mars is –40.0 $^ { \circ } \mathrm { C }$. Using the lapse rate calculated in part (c), find the height in the Martian troposphere at which the temperature is –60.0 $^ { \circ } \mathrm { C }$. (e) Data from the Mariner flights in 1969 indicated a lapse rate in the Martian troposphere of about –1.5$^ { \circ } \mathrm { C }$/km. The Viking missions in 1976 gave measured lapse rates of about –2$^ { \circ } \mathrm { C }$/km. These values deviate from the ideal value calculated in part (c) because of dust in the Martian atmosphere. Why would dust affect the lapse rate? Which mission occurred in dustier conditions, Mariner or Viking? $\frac { d T } { d y } = - \frac { \gamma - 1 } { \gamma } \frac { g M } { R }$

(a) Take the definition of the coefficient of volume expansion to be

$$\beta=\left.\frac{1}{V} \frac{d V}{d T}\right|_{P-\text { constant }}=\frac{1}{V} \frac{\partial V}{\partial T}$$

Use the equation of state for an ideal gas to show that the coefficient of volume expansion for an ideal gas at constant pressure is given by $\beta=1 / T$, where $T$ is the absolute temperature.
(b) What value does this expression predict for $\beta$ at $0^{\circ} \mathrm{C}$ ? State how this result compares with the experimental values for helium and air. Notice that these values are much larger than the coefficients of volume expansion for most liquids and solids.

Solve each equation. Give solutions in exact form. $\log _{3}(x+1)-\log _{3}(x-3)=2$

Review. At 25.0 m below the surface of the sea, where the temperature is 5.00$^ { \circ } \mathrm { C }$, a diver exhales an air bubble having a volume of 1.00 cm3. If the surface temperature of the sea is 20.0$^ { \circ } \mathrm { C }$, what is the volume of the bubble just before it breaks the surface?

The mass of a hot-air balloon and its cargo (not including the air inside) is 200 kg. The air outside is at $10.0^{\circ} \mathrm{C}$ and 101 kPa. The volume of the balloon is $400 \mathrm{m}^{3}$ To what temperature must the air in the balloon be warmed before the balloon will lift off? (Air density at $10.0^{\circ} \mathrm{C}$ is $\left.1.244 \mathrm{kg} / \mathrm{m}^{3} .\right)$

Given that at $20.0^{\circ} \mathrm{C}$, an aluminum ring has an inner diameter of $5.0000 \mathrm{~cm}$ and a brass rod has a diameter of $5.0500 \mathrm{~cm}$. If both are heated together, what temperature must they both reach so that the ring just slips over the rod? Would this latter process work?

Ice has a lower density than liquid water. How does the volume of a kilogram of water change when it freezes to ice?

A small change in the volume of a liquid occurs when heating the liquid at constant pressure. Use the following data to estimate the fractional contribution this change makes to the heat capacity of water between $4.00^{\circ} \text{C}$ and $100^{\circ} \text{C}$. The density of water at $4.00^{\circ} \text{C}$ and $1.00 \text{~atm}$ pressure is $1.000 \text{~g} / \text{cm}^3$. The density of liquid water at $100^{\circ} \text{C}$ and $1.00 \text{~atm}$ pressure is $0.9584 \text{~g} / \text{cm}^3$.

At $20.0^{\circ} \mathrm{C}$, an aluminum ring has an inner diameter of $5.0000 \mathrm{~cm}$ and a brass rod has a diameter of $5.0500 \mathrm{~cm}$. (a) If only the ring is heated, what temperature must it reach so that it will just slip over the rod?

A constant-volume gas thermometer is calibrated in dry ice $\left(-80.0^{\circ} \mathrm{C}\right)$ and in boiling ethyl alcohol $\left(78.0^{\circ} \mathrm{C}\right)$. The respective pressures are $0.900 \mathrm{~atm}$ and $1.635 \mathrm{~atm}$. (a) What value of absolute zero does the calibration yield?

A constant-volume gas thermometer is calibrated in dry ice (that is, carbon dioxide in the solid state, which has temperature of $-80.0^{\circ} \mathrm{C}$ ) and in boiling ethyl alcohol $\left(78.0^{\circ} \mathrm{C}\right)$. The two pressures are $0.900 \mathrm{~atm}$ and $1.635 \mathrm{~atm}$. (a) What Celsius value of absolute zero does the calibration yield? What is the pressure at (b) the freezing point of water and (c) the boiling point of water?

A constant-volume gas thermometer is calibrated in dry ice $\left(-80.0^{\circ} \mathrm{C}\right)$ and in boiling ethyl alcohol $\left(78.0^{\circ} \mathrm{C}\right)$. The respective pressures are $0.900 \mathrm{~atm}$ and $1.635 \mathrm{~atm}$. (b) What pressures would be found at the freezing and boiling points of water? (Note that we have the linear relationship $P=A+B T$, where $A$ and $B$ are constants.)

The cooling rate of a rising air parcel depends on the type of air mass. From the greatest to the least temperature change, rank the following lapse rates for each kilometer of rising air: (a) environmental lapse rate, (b) moist adiabatic lapse rate, (c) dry adiabatic lapse rate.