PC2352 Examples 1

1.) First year revision quiz

Parts 1-5 test the first law, work done on a fluid and properties of an ideal gas. Not all of these will have been met in lectures this year by the time you do this sheet, but they were all covered in first year. As a hint for parts 2 and 3, remember that during a reversible adiabatic process entropy is constant.

Useful links for this quiz are the first law of thermodynamics, the properties of an ideal gas, work (and worked example).

1.) In some process a fluid starts at $P_1$, $V_1$, $T_1$, and ends at $P_2$, $V_2$, $T_2$. Which of the following are true?

a.)

The heat transferred to the system, $Q$, is independent of the path taken (ie depends only on the end points).

b.)

The work done on the system, $W$, is independent of the path taken

c.)

The sum $Q+W$, is independent of the path taken

d.)

The difference $Q-W$, is independent of the path taken

e.)

The change in energy of the fluid is $\frac 3 2 (P_2 V_2-P_1 V_1)$

2.) An ideal monatomic gas is slowly taken from $P_1$, $V_1$, $T_1$, to $P_2$, $V_2$, $T_2$. Match the work done on the gas with the conditions.

a.) Constant pressure	i) $$n R T_1 \ln\left(\frac{V_1}{V_2}\right)$$
b.) Constant volume	ii) $$P_1(V_1-V_2)$$
c.) Constant temperature	iii) $$\frac 3 2 n R(T_2-T_1)$$
d.) Constant entropy	iv) 0

Answer

3.) Match the following expressions for the heat transferred to the system during the process described in question 2 with the conditions.

a.) Constant pressure	$\alpha$) 0
b.) Constant volume	$\beta$) $$-n R T_1 \ln\left(\frac{V_1}{V_2}\right)$$
c.) Constant temperature	$\gamma$) $$-\frac 5 2 P_1(V_1-V_2)$$
d.) Constant entropy	$\delta$) $$\frac 3 2 n R(T_2-T_1)$$

Answer

4.) How well do the following substances approximate to an ideal gas? Rank on a scale of 0 (not at all) to 3 (very well indeed).

a.)

neon at STP

b.)

supercritical CO$_2$

c.)

oxygen at 300$^{\rm o}$K and $10^3$Pa

d.)

water vapour at 450$^{\rm o}$K and 10 atmospheres

e.)

liquid water

Answer

5.) Which of the following applies to all fluids, and which to ideal gases only?

a.)

${}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}W\ge -P{\rm d}V$

b.)

$$W=n R T \ln\left(\frac{V_1}{V_2}\right)$$ for isothermal compression

c.)

${}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}Q=C{\rm d}T$ where $C$ is the appropriate heat capacity

d.)

$PV^\gamma=\hbox{const}$ if ${}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}Q=0$

e.)

$${\rm d}S\ge \frac {{}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}Q} T$$

f.)

${\rm d}E= C_V {\rm d}T$

g.)

$C_P-C_V=n R$

Answer

6.) You have a number of heat reservoirs available, the hottest being at $T_2$ and the coolest being at $T_1$. Which engine (a-d) will you choose for maximum efficiency?

a.) Diesel	$\eta=\displaystyle 1-{1\over\gamma V_1^{\gamma-1}}\left(\frac {V_3^\gamma-V_2^\gamma}{V_3-V_2}\right)$
b.) Joule ideal gas	$\eta=\displaystyle 1- \left({P_1\over P_2}\right)^\frac{\gamma-1}{\gamma}$
c.) Carnot	$\eta=\displaystyle 1-\frac{T_1}{T_2}$
d.) Otto	$\eta=\displaystyle 1-\left({V_2\over V_1}\right)^{\gamma-1}$

Answer

The maximum score is 26. Any mark above 21 is very good. Anything less requires some revision! See the following links: ideal gas, the first law, work, heat engines.