Intermediate Thermodynamics Assignment 1 — Ideal Gas Law  
$\xi$ is a parameter related to your student ID, with $\xi_1$ corresponding to the last digit, $\xi_2$ to the last two digits, $\xi_3$ to the last three digits, etc. For instance, if your ID is 199225962, then $\xi_1=2$, $\xi_2=62$, $\xi_3=962$, $\xi_4=5962$, etc. Keep a copy of the assignment — the assignment will not be handed back to you. You must be capable of remembering the solutions you hand in.
Question #1
Starting from $\vec{F}=m\vec{a}$ applied on a gas particule, show that $$ P=\rho R T=N k_{\rm B} T= \frac{n k_{\rm B} T}{V} = \frac{\overline{R}T}{\overline{v}}$$ with $$ T\equiv \frac{m \overline{q^2}}{3 k_{\rm B}},~~~~R\equiv \frac{k_{\rm B}}{m},~~~~\overline{R}\equiv N_{\rm A} k_{\rm B}$$ Outline the definition of the pressure $P$, the density $\rho$, the number density $N$, and the molar specific volume $\overline{v}$. Note: this question is worth double the points awarded to the other questions.
Question #2
Consider nitrogen gas (N$_2$) at room temperature and atmospheric pressure. Do the following:
(a)  Find the average number of N$_2$ molecules striking the container walls per second per square meter.
(b)  A 1 m$^3$ glass bulb contains N$_2$ gas at a temperature of $300$ K and at a pressure of 1 atmosphere. The glass bulb, which is to be used in conjunction with some other experiment, is itself enclosed in a large evacuated chamber. Unfortunately the glass bulb has, unknown to the experimenter, a small pinhole about $10^{-4}$ cm radius. To assess the importance of this hole, estimate the time required for 1% of the N$_2$ molecules to escape from the bulb into the surrounding vacuum.
Question #3
(a)  If an automobile tire is inflated to $(20+\xi_1)$ psi gauge pressure (i.e, $(20+\xi_1)$ psi above atmospheric pressure) when cold, at 15$^\circ$C, what will the pressure become if the tire is heated to 30$^\circ$C by driving (assuming the tire does not expand further during the driving)?
(b)  What per cent of the air in the tire would then have to be let out in order to reduce the pressure to the original 20 psi, assuming that the process occurs at constant temperature (i.e. isothermal)?
Question #4
(a)  The dirigible Hindenberg had a gas capacity of 200,000 cubic meter. Compare the lifting force at 1 atm and 21$^\circ$C when filled with hydrogen and when filled with helium. (Remember the principle of Archimedes — that buoyancy is equal to the weight of the displaced fluid).
(b)  The Hindenberg could ascend to a maximum altitude of 7,620 meters when filled with hydrogen. The air temperature at this altitude is -18$^\circ$C and the pressure is 1/3 atm; assume that the hydrogen temperature and pressure also have these values. Estimate the weight of the Hindenberg if it were filled with air at 1 atm and placed on a set of scales at sea level.
Question #5
A bottle of gaseous hydrogen cyanide HCN with a molecular weight of 27 gr/mole is moved to a small closed room filled with air. It is known that the volume of the room is of 10 cubic meters, and that the volume of the hydrogen cyanide bottle is of 0.3 cubic meters.
(a)  The pressure in the hydrogen cyanide bottle is set to 1 atm and the temperature to 30$^\circ$C. The air in the room is at 1 atm and 310 K. It is observed that after one hour, the concentration of hydrogen cyanide in the room is one part per million (1 ppm). Calculate the size of the hole in the hydrogen cyanide bottle.
(b)  The experiment of (a) is repeated with the same hydrogen cyanide bottle placed in the same room. You can assume that the hydrogen cyanide and the air both have the same temperature of 40$^\circ$C and that the pressure of the air is of 1 atm. Calculate the minimum pressure inside the hydrogen cyanide bottle that will result in a hydrogen cyanide concentration in air of 100 mg/m$^3$ after 1 day.
Hint: For simplicity, you can assume air to be composed solely of molecular nitrogen.
Due on Wednesday March 20th at 16:30. Do Questions #1, #4, and #5 only.
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