Q: For an adiabatic expansion of an ideal monatomic gas, which of the following relationships holds between pressure and volume?

$$pV^{\gamma}$$ = \text{constant}$$, where $$\gamma = \frac{5}{3}$$

Question 1

A sample of an ideal gas is contained in a cubic box of side $$L as $$shown in the first image. If a molecule of mass $$m$$ and velocity $$v_x$$ collides elastically with the shaded wall, what is the change in momentum of the molecule in the $$x-$$direction after the collision?

Accepted Answer

$$2mv_x$$

Question 2

According to the kinetic theory, the pressure exerted by $$N$$ molecules of an ideal gas in a volume $$V is $$given by which of the following equations?

Accepted Answer

$$p = \frac{mN(v^2$)_{avg}}{3V}$$

Question 3

The root-mean-square speed $$v_{rms} of $$molecules in an ideal gas is given by:

Accepted Answer

$$v_{rms} = \sqrt{\frac{3RT}{M}}$$

Question 4

In the Maxwell speed distribution graph (third image), what does the area under the curve between $$v_1$$ and $$v_2$$ represent?

Accepted Answer

The fraction of molecules with speeds between $$v_1$$ and $$v_2$$

Question 5

For a monatomic ideal gas, what is the average kinetic energy per molecule?

Accepted Answer

$$\frac{3}{2}kT$$

Question 6

A diatomic molecule such as $$O_2 ($$see fourth image) has how many degrees of freedom at room temperature, and what is the corresponding molar specific heat at constant volume $$C_V$$?

Accepted Answer

5 degrees of freedom, $$C_V = \frac{5}{2}R$$

Question 7

The equipartition theorem states that each degree of freedom contributes how much energy per molecule on average?

Accepted Answer

$$\frac{1}{2}kT$$

Question 8

For an adiabatic expansion of an ideal monatomic gas, which of the following relationships holds between pressure and volume?

Accepted Answer

$$pV^{\gamma}$$ = 	ext{constant}$$, where $$\gamma = \frac{5}{3}$$

Question 9

In the $$p$$-$$V$$ diagram for an isothermal process (second image), the work done by the gas as it expands from $$V_i to$$ $$V_f is $$given by:

Accepted Answer

$$W = nRT \ln \frac{V_f}{V_i}$$

Question 10

A sample of hydrogen gas ($$H_2) is $$heated from 100 K to 3000 K. According to the $$C_V/R vs. $$temperature graph (fifth image), what happens to the molar specific heat at constant volume as temperature increases?

Accepted Answer

It increases in steps as more degrees of freedom become accessible

Question 11

During an adiabatic free expansion of an ideal gas (see seventh image), what happens to the temperature of the gas?

Accepted Answer

It remains constant

Question 12

If $$Q$$ joules of heat is added to an ideal gas at constant pressure and the gas does $$W$$ joules of work, what is the change in thermal energy $$\Delta E_{th} of $$the gas?

Accepted Answer

$$\Delta E_{th} = Q - W$$

Question 13

For a polyatomic molecule like $$CH_4 ($$see fourth image), how many rotational degrees of freedom does it have?

Accepted Answer

3

Question 14

The molar specific heat at constant pressure $$C_p$$ for an ideal gas is related to the molar specific heat at constant volume $$C_V by$$:

Accepted Answer

$$C_p = C_V + R$$

Question 15

A sample of an ideal monatomic gas undergoes an isobaric expansion from $$1.0 L to$$ $$4.0 L at a $$pressure of $$32$$ atm. What is the work done by the gas? (Use $$1$$ atm $$= 1.01 	imes 10^5$ Pa, 1$ L $$= $$10^{-3} m$^3)$$

Accepted Answer

$$9.69 	imes 10^3$ J

Question 16

Which process on a $$p$$-$$V$$ diagram corresponds to a constant volume process?

Accepted Answer

A vertical line

Question 17

If the molar mass $$M of a $$gas is doubled while keeping temperature constant, what happens to the root-mean-square speed $$v_{rms}$$?

Accepted Answer

It decreases by a factor of $$\sqrt{2}$$

Question 18

For a diatomic ideal gas, what is the total thermal energy $$E_{th}$$ for $$n$$ moles at temperature $$T$$?

Accepted Answer

$$E_{th} = \frac{5}{2} nRT$$

Question 19

A gas undergoes an isothermal expansion from $$V_i to$$ $$V_f. If$$ $$V_f \u003e V_i$$, what can be said about the work done by the gas?

Accepted Answer

The work is positive

Question 20

Which of the following is true for an adiabatic process in an ideal gas?

Accepted Answer

No heat is transferred to or from the gas ($$Q = 0)$$

Kinetic Theory and Thermodynamics of Ideal Gases

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