2.E: Gases (Exercises)
- Page ID
- 84446
Q2.1
Assuming the form of the Maxwell distribution allowing for motion in three directions to be
\[ f(v) = N v^2 \text{exp} \left( \dfrac{m v^2}{2 k_BT} \right) \label{MBFullN}\]
derive the correct expression for \(N\) such that the distribution is normalized. Hint: a table of definite integrals indicates
\[ \int_0^{\infty} x^{2n} e^{-ax^2} dx = \dfrac{1}{4} \dfrac{\sqrt{\pi}}{a^{3/2}}\]
Q2.2
Dry ice (solid CO_{2}) has a density of 1.6 g/cm^{3}. Assuming spherical molecules, estimate the collisional cross section for \(CO_2\). How does it compare to the value listed in the text?
Q2.3
Calculate the pressure exerted by 1.00 mol of Ar, N_{2}, and CO_{2} as an ideal gas, a van der Waals gas, and a Redlich-Kwong gas, at 25 °C and 24.4 L.
Q2.4
The compression factor \(Z\) for CO_{2} at 0 °C and 100 atm is 0.2007. Calculate the volume of a 2.50 mole sample of CO_{2} at 0 °C and 100 atm.
Q2.5
\(Ar\) | \(N_2\) | \(CO_2\) | |
---|---|---|---|
ideal | |||
van der Waals | |||
Redlich-Kwong |
Q2.6
What is the maximum pressure that will afford a N_{2} molecule a mean-free-path of at least 1.00 m at 25 °C?
Q2.7
In a Knudsen cell, the effusion orifice is measured to be 0.50 mm^{2}. If a sample of naphthalene is allowed to effuse for 1.0 hr at a temperature of 40.3 °C, the cell loses 0.0236 g. From this data, calculate the vapor pressure of naphthalene at this temperature.
Q2.8
The vapor pressure of scandium was determined using a Knudsen cell [Kirkorian, J. Phys. Chem., 67, 1586 (1963)]. The data from the experiment are given below.
Vapor Pressure of Scandium | |
---|---|
Temperature | 1555.4 K |
Time | 110.5 min |
Mass loss | 9.57 mg |
Diameter of orifice | 0.2965 cm |
From this data, find the vapor pressure of scandium at 1555.4 K.
Q2.9
A thermalized sample of gas is one that has a distribution of molecular speeds given by the Maxwell-Boltzmann distribution. Considering a sample of N_{2} at 25 ^{c}C what fraction of the molecules have a speed less than
- the most probably speed
- the average sped
- the RMS speed?
- The RMS speed of helium atoms under the same conditions?
Q2.10
Assume that a person has a body surface area of 2.0 m^{2}. Calculate the number of collisions per second with the total surface area of this person at 25 °C and 1.00 atm. (For convenience, assume air is 100% N_{2})
Q2.11
Two identical balloons are inflated to a volume of 1.00 L with a particular gas. After 12 hours, the volume of one balloon has decreased by 0.200 L. In the same time, the volume of the other balloon has decreased by 0.0603 L. If the lighter of the two gases was helium, what is the molar mass of the heavier gas?
Q2.12
Assuming it is a van der Waals gas, calculate the critical temperature, pressure and volume for \(CO_2\).
Q2.13
Find an expression in terms of van der Waals coefficients for the Boyle temperature. (Hint: use the viral expansion of the van der Waals equation to find an expression for the second viral coefficient!)
Q2.14
Consider a gas that follows the equation of state
\[p =\dfrac{RT}{V_m - b}\]
Using a virial expansion, find an expression for the second virial coefficient.
Q2.15
Consider a gas that obeys the equation of state
\[ p =\dfrac{nRT}{V_m - b}\]
where a and b are non-zero constants. Does this gas exhibit critical behavior? If so, find expressions for \(p_c\), \(V_c\), and \(T_c\) in terms of the constants \(a\), \(b\), and \(R\).
Q2.16
Consider a gas that obeys the equation of state
\[ p = \dfrac{nRT}{V- nB}-\dfrac{an}{V}\]
- Find an expression for the Boyle temperature in terms of the constant \(a\), \(b\), and \(R\).
- Does this gas exhibit critical behavior? If so, find expressions for \(p_c\), \(V_c\), and \(T_c\) in terms of the constants \(a\), \(b\), and \(R\).