Chapter 19: The Kinetic Theory of Gases

A steel tank contains$\mathbf{300}\mathbf{}\mathbf{g}\mathbf{}$ of ammonia gas (${\mathbf{NH}}_3$ ) at a pressure of$\mathbf{1}\mathbf{.}\mathbf{35}\mathbf{\times }{\mathbf{10}}^6\mathbf{pa}$ and a temperature of $\mathbf{77}°\mathbf{C}$.

(a) What is the volume of the tank in liters?

(b) Later the temperature is $\mathbf{22}\mathbf{°}\mathbf{C}$and the pressure is$\mathbf{8}\mathbf{.}\mathbf{7}\mathbf{\times }{\mathbf{10}}^5\mathbf{}\mathbf{pa}$ . How many grams of gas have leaked out of the tank?

In an industrial process the volume of$\mathbf{25}\mathbf{.0}\mathbf{\text{\hspace{0.17em}mol}}\mathbf{}$of a monatomic ideal gas is reduced at a uniform rate from$\mathbf{0}\mathbf{.616}{\mathbf{\text{\hspace{0.17em}m}}}^{\mathbf{3}}$to$\mathbf{0}\mathbf{.308}{\mathbf{\text{\hspace{0.17em}m}}}^{\mathbf{3}}$in$\mathbf{2}\mathbf{.00}\mathbf{\text{h}}\mathbf{}$while its temperature is increased at a uniform rate from$\mathbf{27}\mathbf{.0}\mathbf{°}\mathbf{\text{C}}$to$\mathbf{450}\mathbf{°}\mathbf{\text{C}}$. Throughout the process, the gas passes through thermodynamic equilibrium states. What are

(a) the cumulative work done on the gas,

(b) the cumulative energy absorbed by the gas as heat, and

(c) the molar specific heat for the process? (Hint: To evaluate the integral for the work, you might use$\mathbf{\int }\frac{\mathbf{a}\mathbf{+}\mathbf{b}\mathbf{x}}{\mathbf{A}\mathbf{+}\mathbf{B}\mathbf{x}}\mathit{d}\mathit{x}\mathbf{=}\frac{\mathbf{b}\mathbf{x}}{\mathbf{B}}\mathbf{+}\frac{\mathbf{a}\mathbf{B}\mathbf{-}\mathbf{b}\mathbf{A}}{{\mathbf{B}}^{\mathbf{2}}}\mathit{I}\mathit{n}\mathbf{(}\mathit{A}\mathbf{+}\mathit{B}\mathit{x}\mathbf{)}$an indefinite integral.) Suppose the process is replaced with a two-step process that reaches the same final state. In step 1, the gas volume is reduced at constant temperature, and in step 2 the temperature is increased at constant volume. For this process, what are

(d) the cumulative work done on the gas,

(e) the cumulative energy absorbed by the gas as heat,and

(f) the molar specific heat for the process?

Question: Figureshows a cycle consisting of five paths: AB is isothermal at 300K, BC is adiabatic with $\text{work}=5.0\text{J}$, CD is at a constant pressure of, DE is isothermal, and EA is adiabatic with a change in internal energy of $8.0\text{J}$. What is the change in internal energy of the gas along path CD?

Question: A pipe of length L = 25. 0 m that is open at one end contains air at atmospheric pressure. It is thrust vertically into a freshwater lake until the water rises halfway up in the pipe (Fig.). What is the depth h of the lower end of the pipe? Assume that the temperature is the same everywhere and does not change.

Compute

1. The number of moles
2. The number of molecules in$\mathbf{1}\mathbf{.}\mathbf{00}{\mathbf{cm}}^3$of an ideal gas at a pressure of$\mathbf{100}\mathbf{Pa}$and a temperature of$\mathbf{220}\mathbf{K}$.

The dot in Fig $\mathbf{19}\mathbf{-}\mathbf{18}\mathit{c}$represents the initial state of a gas, and the adiabatic through the dot divides the p-V diagram into regions $\mathbf{1}$and $\mathbf{2}$. For the following processes, determine whether the corresponding heat is positive, negative, or zero: (a) the gas moves up along the adiabatic, (b) it moves down along the adiabatic, (c) it moves to anywhere in region $\mathbf{1}$, and (d) it moves to anywhere in region $\mathbf{2}$.

In a motorcycle engine, a piston is forced down toward the crankshaft when the fuel in the top of the piston’s cylinder undergoes combustion. The mixture of gaseous combustion products then expands adiabatically as the piston descends. Find the average power in (a) watts and (b) horsepower that is involved in this expansion when the engine is running at 4000rpm, assuming that the gauge pressure immediately after combustion is 15atm, the initial volume is 50cm³, and the volume of the mixture at the bottom of the stroke is 250cm³. Assume that the gases are diatomic and that the time involved in the expansion is one-half that of the total cycle.

For adiabatic processes in an ideal gas, show that (a) the bulk modulus is given bywhere(See Eq. 17-2.) (b) Then show that the speed of sound in the gas is${\mathbf{v}}_{\mathbf{s}}\mathbf{=}\sqrt{\frac{\mathbf{\gamma p}}{\mathbf{\rho }}}\mathbf{=}\sqrt{\frac{\mathbf{\gamma RT}}{\mathbf{M}}}\mathbf{,}$where is the density, T is the temperature, and M is the molar mass. (See Eq. 17-3.)

Question: Air at 0.00 ⁰C and 1 atm pressure has a density of $\mathbf{1}\mathbf{.}\mathbf{29}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{3}\mathbf{}}\mathbf{}\mathit{g}\mathbf{/}\mathit{c}{\mathit{m}}^{\mathbf{3}}$, and the speed of sound is 331 m/s at that temperature. Compute the ratio $\lambda$of the molar specific heats of air. (Hint: See Problem 91)

Question: The speed of sound in different gases at a certain temperature T depends on the molar mass of the gases. Show that $\frac{{\mathbf{v}}_{\mathbf{1}}}{{\mathbf{v}}_{\mathbf{2}}}\mathbf{=}\sqrt{\frac{{\mathbf{M}}_{\mathbf{2}}}{{\mathbf{M}}_{\mathbf{1}}}}$whereis the speed of sound in a gas of molar mass M₁and v₂is the speed of sound in a gas of molar mass M₂.(Hint: See Problem 91.)